In the following answer I'll try to:
- explain the ahadith quoted.
- post some commentaries and give my own comment (maybe a bit scattered here and there).
- show that there are narrations that seem to contradict them.
- post based on other narration an explanation of the narrator of one of the two quoted hadith himself and how he understood the matter.
An explanation and a first overview on the ahadith
Both ahadith say that writing down hadith was or is not allowed!
The first hadith is also emphasizing on only telling the truth about Muhammad and warning those who attributed any falsehood! In fact allowing to tell or to narrate hadith. So an oral transmission is allowed and this goes along with Qur'an. As it clearly orders the mothers of believers to tell or transmit what they hear in their houses, no matter if it is Qur'an or hadith:
And remember what is recited in your houses of the verses of Allah and wisdom. Indeed, Allah is ever Subtle and Acquainted [with all things]. (33:34)
And this is what they did and what other sahaba did as they understood this verse as an order to anybody who witnessed the Prophet.
Note that without this order we wouldn't know much about the personality of Muhammad nor of him as a father, husband etc.
So saying that the two hadith you quoted are discrediting all other hadith on this basis already is wrong! We could discuss about hadith compilations or collections in written form but not on hadith itself!
Now we also need to take a look at the both narrators:
Zaid ibn Thabet is known as a scribe of revelation and was young man when the Prophet died.
Abu Sa'id al-Khudri is known to have been a too young to assist at the battle of Uhud but 2 years later was allowed to participate at the battle of al-Khandaq one therefore could assume he then was around 14 or 15 based on the answer of this question: Does Islam allow child soldiers?. But one must be aware that at the battle of al-Khandaq or al-Ahzaab as it is also known Medina was besieged, so maybe this was a special circumstance! So it is hard to tell his age then and maybe it doesn't play a role here.
Let's take a look at the hadith compilations: Imam an-Nawawi (as it is him who entitled or divided the sahih Muslim into chapters) compiled the hadith of abu Sa'id al-Khudri in the Chapter "The Book of Zuhd and Softening of Hearts" in the section called " Chapter: Verification Of Hadith And The Ruling On Writing Down Knowledge" and it is not the first hadith in the section while the first hadith more or less goes along with the last statement of this hadith, which you apparently are not interested in "he who attributed any falsehood to me-and Hammam said: I think he also said: "deliberately"-he should in fact find his abode in the Hell-Fire.".
Abu Dawod compiled the hadith of Zaid ibn Thabet in the chapter called "Book of Knowledge (Kitab Al-Ilm)" in a section entitled "Writing knowledge" and here again it is not the first hadith there. The first hadith in this section is even a contradiction to this hadith! This special section includes 5 ahadith from which some seem to allow writing hadith while other don't allow it. I'd like to point here at a hadith of abu Sa'id al-Khudri which emphasizes his statement from the hadith in sahih Muslim to some extent.
I'd like to point at the fact that even if Qur'an was already written on pieces of what ever came in hand it was not collected as a book at the time of our Prophet and compiling it in a book was not an easy matter, as people were aware that what they might be doing is something the Prophet himself never did, so it needed a discussions and argumentations to realize it!
Commentaries on the ahadith
The hadith includes in the narrator chains lots of more or less untrustworthy or weak narrators it's therefore qualified as da'if and in the hadith commentary I could read so far they make a link to the hadith you quoted from sahih Muslim.
The narrators which make it weak are:
Abu Ahmad az-Zubairi أبو أحمد الزبيري is considered as reliable but not if he narrates from at-Thawri. (See also)
Kathir ibn Zaid كثير بن زيد الأسلمي is reliable but made mistakes
Al-Muttalib ibn Abdullah ibn Hantab المطلب بن عبد الله بن حنطب used to do maraseel, some say he used to narrate on the authority of 'Aisha and doesn't seem to have met her. (See also)
IMO -and I'm neither a specialist or a scholar- but I can't see why this particular chain is considered as da'if by al-Albani (I'm not aware of other qualifications) and I'd like to add that he qualified the hadith mentioned above as shadh! Please also be aware that this is a marfo' hadith.
Note that in the narration chain of the hadith from sahih Muslim all of the narrators are widely accepted Imam ad-Dhahabi said I don't know the reason for an-Nasa'i to qualify Hudab ibn Khalid هداب بن خالد الأزدي (who is the teacher of Muslim or the narrator Muslim got this hadith from) as da'if!
So scholars gave explanations like following:
Al-Nawawi said in his commentary on Saheeh Muslim:
“Al-Qaadi said: there were many disputes among the Sahaabah and Taabi’een concerning the writing down of knowledge. Many of them regarded this as being makrooh, but most of them regarded it as permissible. This dispute is no longer an issue.
They differed as to the meaning of this hadeeth which says that it is forbidden. It was said that this pertained to one who was sure of his memory, and there was the fear that he may rely upon what he had written if he wrote it down; the ahaadeeth which say that it is permissible to write things down is to be interpreted as referring to the one whose memory is not reliable, such as the hadeeth, “Write it down for Abu Shaah”; or the hadeeth of the saheefah of ‘Ali (may Allaah be pleased with him); the hadeeth of the book of ‘Amr ibn Hazm, which contains laws on inheritance, sunnahs and diyaat (blood money); the hadeeth about writing down charity, and the minimum amounts at which zakaah becomes obligatory (nisaab), with which Abu Bakr sent Anas (may Allaah be pleased with him) to Bahrain; the hadeeth of Abu Hurayrah which says that Ibn ‘Amr ibn al-‘Aas used to ; write things down but he (Abu Hurayrah) did not write things down, and other ahaadeeth. And it was said that the hadeeth forbidding writing down ahaadeeth was abrogated by these ahaadeeth. The prohibition was in effect when there was the fear that (the words of the Prophet (peace and blessings of Allaah be upon him) might be mixed with the Qur’aan. When that danger was no longer present, permission was given to write down (ahaadeeth). And it was said that the prohibition mentioned in the hadeeth referred to writing ahaadeeth on the same page as Qur’aan, lest they become mixed and thus the reader would be confused when looking at this page. And Allaah knows best. (Quote form the fatwa from islamqa in my source list)
I assume an-Nawawi is referring al-Qadi 'Iyad. And this quote already indicates a lot of ahadith which seem to allow writing down hadith.
Other possible reasons for a prohibition could be:
Some explain those hadiths as follows: they should make a clear cut between revelation and hadith. So one could understand it as when there's a revelation don't write down anything else! As there are ahadith indicating that some of the scribes were listening and writing not only on times of a new revelation. So it is narrated that in Ubay ibn Ka'abs private mushaf he added what some may know as du'a al-Qunoot "Allahuma inna nast'eenu bika wa nastaghfiruka ..." as a surah. And mixing Quran the words of Allah with other things or hiding parts of it or adding to it is a serious matter as we know from Quran like one can read in verses like 2:42, 2:74,2:79 and 3:187.
the hadith addresses people who were close to idol worship. They beside of this were (to some extent: 'Omar was not allowed to copy and keep copies of the thorah, while Zaid ibn Thabet even was asked to learn Hebrew to be able to read it) frowned upon to read in the books of people of the book or copy them.
Also it might have been a necessary command as the Qur'an at the time was not yet complete and maybe some verses were abrogated and if we think of the later collection of the moshaf, this was a big help for Zaid ibn Thabet and others!
Beside the confusion issue the words of Allah should be treated with as respectful as possible, so they shouldn't get mixed with the words of a human and the Prophet has chosen to have them written down to make them fix. While the words or doings of the Prophet are witnessed and have a lower value, so writing them down would make the word of Allah equal to the word of the Prophet and again as they were close to Idolatry this would make the matter risky! Malik narrated a statement of 'Omar -who at first wanted to compile a hadith book- saying: "There shouldn't be a book beside the book of Allah!"
Ibn Hajar also added that this prohibition was temporary: that means it was prohibited to the scribes to write while they were listening to a new revelation. (I'd like to know whether he counts abu Sa'id al-Khudri among the revelation scribes?)
Also note the fact that possibilities to write were not given (they didn't have paper or enough material) so they used bones, pieces of wood or palmed stalks or thin white stones ... in short what ever could be used to be written on. If they did the same for hadith they might have a problem!
I'd like also to point at a matter I've experienced: It is hard to memorize two things at once. And it helps a lot to write things down to memorize them and to read them from your own transcription. I've even been told by a hadfidh of Quran if one wants to memorize it based on a moshaf it is necessary to always use the same printed edition (or at best always the same copy).
Some sahaba were even allowed to write down hadith
On the other hand we have ahadith indicating or clearly contradicting this like this hadith:
I used to write everything which I heard from the Messenger of Allah (ﷺ). I intended (by it) to memorise it. The Quraysh prohibited me saying: Do you write everything that you hear from him while the Messenger of Allah (ﷺ) is a human being: he speaks in anger and pleasure? So I stopped writing, and mentioned it to the Messenger of Allah (ﷺ). He signalled with his finger to his mouth and said: Write, by Him in Whose hand my soul lies, only right comes out from it. (Sunan abi Dawod)
The fact that Abdullah ibn 'Amr ibn al-'Aas used to write is also confirmed by abu Hurairah who converted two or at most three years before the death of the Prophet (very close to the time of the end of revelation) in a this narration:
There is none among the companions of the Prophet (ﷺ) who has narrated more Hadiths than I except 'Abdullah bin 'Amr (bin Al-'As) who used to write them and I never did the same. (Sahih al-Bukhari, Jami' at-Tirmdihi)
The version of Imam Ahmad in his Musnad emphasizes the fact that Abdullah used to record hadith, and
ما كان أحفظ لحديث رسول الله - صلى الله عليه وسلم - مني إلا عبدالله بن عمرو؛ فإنه كان يعي بقلبه وكنتُ أعي بقلبي، وكان يكتب وأنا لا أكتب؛ استأذن رسول الله في ذلك فأذِن له
Due to a problem with my internet connection I've lost an even stronger statement I found during my research I hope I can find it!
It was narrated that the collection of Abdullah ibn 'Amr was entitled "as-Sadiqah" and it was later narrated by his grandson 'Amr ibn Shu'aib, also other sahaba have been known for writing down hadith among them Anas ibn Malik and Jabir ibn Abdullah. I'd like to add that 'abdullah ibn 'amr ibn al-'Aas was not among the scribes of revelation as far as I could tell after a certain research however his father and some of his brothers etc. were (See for example on wikipedia, some say they are even less than wikipedia counts)!
An other narration from sahih al-Bukhari shows that our Prophet even allowed to write a speech of the Prophet to somebody (you could read a short commentary on it here in Arabic or go back to the quote from the commentary of an-Nawai on sahih Muslim)! An explanation for the this might be that this man (abu Shah) had a bad memory.
All of this show that the prohibition was not general or at least only a temporary matter!
Abu Sa'id al-Khudry explains how he interpreted the prohibition
To explain your 2nd quoted hadith I use a narration quoting the explanation of the narrator of the 1st (Abu Sa'id al-Khudry) you can read in sunan ad-Darimi (one of the teachers of al-Bukhari):
عَنْ أَبِي نَضْرَةَ ، قَالَ : قُلْتُ لِأَبِي سَعِيدٍ الْخُدْرِيِّ رَضِيَ اللهُ عَنْهُ : أَلَا تُكْتِبُنَا ، فَإِنَّا لَا نَحْفَظُ ؟ ، فَقَالَ : " لَا ، إِنَّا لَنْ نُكْتِبَكُمْ ، وَلَنْ نَجْعَلَهُ قُرْآنًا ، وَلَكِنْ احْفَظُوا عَنَّا كَمَا حَفِظْنَا نَحْنُ عَنْ رَسُولِ اللَّهِ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ
---(My own translation and interpretation take it with care)---
... From abu Nadarah, who said: I said to abu Sa'id al-Khudry (May Allah be pleased with him): "Why don't you let us write as we are not (or able to memorize) memorizing?", He answered: "No, we don't let you write, and we won't let it be (similar to) Quran (Maybe he mean fixed as a revelation), but you should memorize from us like we memorized it from the Messenger of Allah (Peace be upon him).
So abu Sa'id al-Khudri prefred memorizing hadith to writting it!
Conclusion
From all of this we can say spreading hadith is allowed, but one should be careful! Therefore 'Aisha in the hadith which is compiled in the same chapter as the one you quoted warned abu Hurairah from telling too much hadith at once, as this could create confusion to the listening audience!
Scholars and sahaba always prefered memorizing hadith. Therefore until now people are narrating hadith based on a narrator chain back to the source. Imam an-Nawawi is known to have memorized and collected the hadith of almost all major books of hadith with a high sanad (a short narrator chain). I myself know people who have a sanad to Imam an-Nawawi on some ahadith. The hadith books are for those people the same as the mushafs for people who memorize the Quran only a helping tool (for revision etc.). While for us layman a book is more useful to get to know about hadith and about Islam and even start memorizing etc..
Note that for Muwatta' Malik we know a couple of narrators of the book -who were students of Imam Malik- like al-Qa'nabi, Yahya ibn Yahya al-Laithi, Yahya ibn Yahya ibn Kathir, Abdullah ibn Wahb and many others but somehow apparently no body among us layman knows that sahih al-Bukhari has been transmitted to us via narrators too like abu Ishaaq al-Balkhi, Muhammad ibn Yoosuf al-Farbari (died 320 A.H.) and an-Nafasi (died 295 A.H.) (for more narrators and even narrator chains of sahih al-Bukhari until the 6th century take a look at this post and this fatwa).
Therefore it was long after the sahaba still discussed whether hadith should be collected or written down or not, even if there are records for it already being written in the time of Prophet, by Abdullah ibn Amr ibn al-'Aas, 'Ali ibn abi Talib and others etc. beside letters which might include something else then Quran (maybe hadith). Many sahaba still emphasized on memorizing the hadith via oral transmission among them were sahba like Abu Sa'id al-Khudri, abu Hurairah maybe because of their respect for Quran, or their fear of telling falsehood, or what they found the best way to preserve hadith (as Quran was also orally transmitted) or they just followed what the understood as an order of the Prophet etc..
At least at a certain time the Muslim nation could find a consensus to allow preserving hadith also in books like it was done with the Quran. But to be safe from telling falsehood they might still attribute ahadith to the people who collected or compiled it. One of the first who gave order to collect hadith for this purpose was 'Omar ibn 'Abdal'aziz (and I'm quite not sure whether at the time he was yet called Caliph or only waly al-Medina).
For details you could look at articles like (mostly Arabic sources):
Here on islamweb.
A detailed article with many references.
And this fatwa answering a question about the correctness of the hadith you quoted 1st with explanations.
