My answer relies a lot on the deleted answer of @Medi1Saif:
Based on ibn 'Ashirs عبد الواحد بن عاشر
definition in his al-Murshid al-Mo'yn المرشد المعين على الضروري من علوم الدين (A Maliki reference):
مُقدِمةُ مِنَ الأُصُولِ
مُعِينَةُ في فُروعِهَا عَلَى الوُصُولِ
الْحُكْمُ في الشَّرْعِ خِطابُ رَبِّنَا ***
الْمُقْتضِي فِعْلَ الْمُكَلَّفِ افْطُنَا
بِطِلَبِ أوْ إِذْنِ أوْ بِوَضْعِ ***
لِسَبَبِ أَوْ شَرْطِ أوْ ذِي مَنْعِ
أَقْسامُ حُكْمِ الشَّرْعِ خَمْسّةُ تُرامْ***
فَرْضٌ ونَدْبٌ وكَراهَةٌ حَرامْ
ثُمَّ إِبَاحَةُ فَمَأْمُورُ جُزِمْ ***
فَرْضُ ودونَ الْجَزْمِ مَنْدوبُ وُسِم (source or here)
which was translated in English as follows (source):
(Introductory Chapter About the Foundational Principles of Jurisprudence
Which Help One Reach Understanding of its Branches)
A ruling in the Shari`ah is the speech of our Lord
Comment:(read in 12:40 " Legislation is not but for Allah." which is used as the strongest evidence for that.)
| which determines (the rank) of the action of the person
who is responsible (for his actions). So, understand this well.
(This ruling is either) a request, a permission, or a stipulation - | with a triggering cause, precondition, or that of
prevention.
The divisions of legal rulings are five desired: | (1) obligatory, (2) recommended, (3) disliked, (4) unlawful, And lastly (5) neutral and allowed. So the command which is strict | is called a fard!/wājib. And if it is of a lower level
of strictness, it is marked as mandūb.
with the definitions of the terms:
The next five legal ruling terms form the basic vocabulary of Jurisprudence. One cannot study, understand,
or follow Law without the knowledge of what these five terms mean:
The first term is wājib, which in (Arabic) language means obligatory.
The second term is mandūb, which in language means recommended.
The third term is mubāh, which in language means neutral and allowed.
The fourth term is makrūh, which in language means disliked.
The fifth term is harām, which in language means unlawful.
...
Wajib is the legal ruling for actions that Allah has commanded us to perform and not given us a choice to leave undone.
Mandub is the legal ruling for actions that Allah has recommended that we perform, but He has given us a choice to leave them undone.
So wajib is anything which we may get rewards if we do it and sin if we leave it. As Allah says:
... So let those beware who dissent from the Prophet's order, lest fitnah strike them or a painful punishment. (24:63)
While for Mandub we have an option: We may get a reward if we perform it, but won't be sinning if we leave it.
The keyword for what constitutes fard is "الْجَزْمِ" which means (determination) or no single doubt or choice!
Note that this is also the definition of the Hanbali school of fiqh and covers the definition of the Shafi'i school of fiqh which adds two other kinds of legal rulings sahih (true, correct) and batil (incorrect, false) while the Hanafis make a distinction between fard and wajib (which basically is that fard is based on a clear order, while wajib is based on a concluded order). For distinguishing orders some scholars point at the importance of verbs in the order or imperative form: افعل.
You may also read for example from in al-Waraqaat (pdf- page 56-57) of imam al-Juwayni the following definition of an order or command:
والأمر: استدعاء الفعل بالقول، ممن هو دونه، على سبيل الوجوب
وصيغته : افعل،
وهي ثم الإطلاق والتجرد عن القرينة تحمل عليه، إلا ما دل الدليل على أن المراد منه الندب، أو الإباحة،
ولا تقتضي التكرار على الصحيح، إلا ما دل الدليل على قصد التكرار، ولا تقتضى الفور
والأمر بإيجاد الفعل أمر به، وبما لا يتم الفعل إلا به، كالأمر بالصلاة؛ فإنه أمر بالطهارة؛ المؤدية إليها، وإذا فُعِل يخرج المأمور عن العهدة
A command is a verbal demand obliging an inferior (someone who is below) to
perform an act. The verbal form that indicates command is if'al [ افعل - the
imperative].
When unqualified, and in the absence of contextual indications [to the
contrary], it is interpreted as obligation, except when some evidence indicates
that recommendation (الندب) or permission (الإباحة) is meant, in which case it is interpreted accordingly.
The correct view is that command does not require the repetition of the act,
unless some evidence indicates that repetition was intended; nor does it require
immediate action.
The command to bring about an action is a command to perform both the act
and whatever is required for the completion of the act, just as the command to
perform the prayers is a command enjoining the purity that paves the way for
them. When the commanded act is performed, the one who is commanded is
considered to have discharged his responsibility.
As an example you may take:
- ... So recite what is easy from it and establish prayer and give zakah ... (73:20)
All three: recite, establish and give are expressing an order which make no doubt!
Now if the order is less strict you move to recommended like:
- O you who have believed, when you contract a debt for a specified term, write it down. ... 2:282
Check in Tafsir al-Qurtobi his discussion of the ruling: The majority is of the opinion that writing a contract in this situation is only mandub.
- ... then make a contract with them if you know there is within them goodness ... (24:33)
The ruling is the same as above writing a contract is mandub and not necessary. According to the commentary on a poem -written by al-'Amrity العَمْرِيطِي- based on al-Juwayni's al-Warqat of Muhammad Alawi al-Maliki.
- ... Then when you release their property to them, bring witnesses upon them. ... (4:6)
and from the sunnah:
- Allah's Messenger (ﷺ) said, "If I had not found it hard for my followers or the people, I would have ordered them to clean their teeth with Siwak for every prayer." (sahih al-Bukhari)
- The Prophet (ﷺ) said, "Pray before the Maghrib (compulsory) prayer." He (said it thrice) and in the third time, he said, "Whoever wants to offer it can do so." He said so because he did not like the people to take it as a tradition. (sahih al-Bukhari)
My understanding of mandub is that the difference between mandub and wajib is that there might be a doubt, an optiona or a specific condition or a difference in interpretation/opinion or even a kind of contradiction with other evidences. All this creates a kind of (minor) doubt, while an order must be crystal clear!
Some of the sources or osol which declare fard are the Qur'an (from Allah), sunnah (from the Prophet whom is declared to speak the truth by Allah) and Ijma' (is supported by Qur'an and sunnah as a source determinate of knowledge) as these souces give what we may call a determinate evidence, while one may consider analogy rather human (and failible) source. Also note that the first three sources are those who are common between all madhhabs even if there might be a difference in interpretation and definition.
Discussion of some of your examples:
I've tried to address them according my understanding and Allah knows best.
•To make ghusl in fridays (not fard):
This can hardly be considered as fard if one believe that the hadith in sahih Muslim quoting wudu' instead of ghusl is sahih as there would be a contradiction. Therefore considering it as strongly recommended is the best was out!
•To read the Quran with tajweed (many say it is fard if you know how to):
Here we should know that memorizing the Qur'an is fard kifaya and therefore reciting it correctly also should be the same, as one shouldn't memorize Qur'an without knowing the correct tajweed! Beside this not all Muslims are fluent Arabic speakers! Also read Can someone give me examples of fard kifaya?
•To pray Friday prayer/Jumu'a (for men), (fard):
Well when you read the Arabic text one may say hey way do Muslims make here a distinction between women and men (which usually is not the case in worship matters), but we have the sunnah and ijma' who tell us that this is the way it has been performed. Also note that the verb expressing the order here is in passive form so it is not a strict order in Arabic language and the sentence is also conditional due to the term: "when is called". All this might explain a restriction. Note that fard kifaya has strong similarities to sunnah/mandub as one must not do it and wouldn't be sinning if only one Muslim performed it, but if anybody else left it also all would sin
•To only eat meat where the name of God has been provoked. (disagreements, many say it is not fard):
and
•To say Bismillah over food (not fard):
These both are due to the verse from surat al-Maidah allowing Muslims to eat the food of the people of the book. So one could speak about a kind of naskh in this case.
•To grow a beard (for men), (fard according to most, while disagreements exists):
Here there are some athar saying that ibn 'Omar -one of the sahaba whom followed the sunnah one could say extensively- used to take from his beard (shorten it) which a few scholars interpret as an indication to allow shaving it.
Some references I've consulted:
islamqa, al-mustasfa of al-Ghazaly (shafi'i), sharh mukhtassar ar-Rawdah also known as al-Bulbul of at-Toufi الطوفي -not at-Toussi as quoted by islamweb- (hanbali)
Note: When I started investigating I found that in some of the examples you presented we may find the opinion of one (maybe two) single scholars against the majority and I've read in tafsir al-Qurtobi a comment on one point where at-Tabari held an opinion which was hardly supported by any source while the majority held was in consensus about a different ruling, al-Qurtobi commented this that saying something like "at-Tabari held the wrong opinion that..." .Unfortunately I couldn't find that comment nor recall the verse I've been checking at the time.
