A bit off topic but maybe a necessary intro:
Conditions for an accepted Reading
Any reading (Qira-a) or recitation which fulfills the following 3 conditions is considered to be sahih (sane) (be aware that they may conflict each other in some cases):
the recitation has an authentic chain of narration in which the chain of narrators was continuous until the Prophet Muhammed (peace be upon him); the narrators were all known to be righteous and they were all known to possess good memories. It was also required that the recitation be conveyed by a large number of narrators on each level of the chain of narration below the level of Sahabah (the condition of Tawaatur).
the variations in recitations match known Arabic grammatical constructions:
the recitation to coincide with the script of one of the copies of the Quran distributed during the era of Caliph 'Othmân is correct.
This applies for all ten Qira'at which are known these days and all the riwaya'at which follow these Qira'at as they have the highest degree of tawatur and match known Arabic grammatical constructions and coincide with the script of the copies of Quran distributed during the era of Caliph 'Othman ibn 'Affan (May Allah be pleased with him) with rather few differences. More details could be found in What are the readings (qira'at) of Quran? and here about differences in the copies of those distributed copies.
Differences in the recitation of verse 6:63
In riwayat Hafs (and all Qira'at of al Kufa) you'll find
لَئِنْ أَنْجَانَا (la-in anjaana): Meaning if he should save us...
While in riwayat Warsh (and in all non-Kufi Qira'at) you'll find
لَئِنْ أَنْجَيْتَنَا: Meaning if You saved us ...
in the typical form of Warsh لَئِنَ انْجَيْتَنَا (la-in-anjyatana) where the sound of the alif (a hamza would appear in riwayat Qalun) "al fatha" is transported to the letter before it, so that both of them would have "a fatha" and the alif would be written without hamza!
This recitation is a direct speech instead of the passive form from riwayat hafs!
So there's a clear Arabic meaning and a tawatur as the Qira'at of Imam Nafi' is maybe the one with the highest degree of tawatur!
While
انجــينا (anjyna or anjay-na) which at least doesn't make sense in this context
Note that the different qira'at and riwayat hold different rulings of tajweed and recitation of the Qur'an so لَئِنْ أَنْجَانَا in Hafs wouldn't be performed the same way as in ad-Dury 'an al-Kissaai for example.
So far we have only considered one out of many sciences of the Qur'an the science of the qira'at and neglected another important science the science of rasm al-Qur'an in which imam ad-Dani أبو عمرو الداني (a specialist in many Qur'an sciences and author of many great books in different topics of the Qur'an sciences) had leadership especially with his book al-Muqni' المقنع في رسم مصاحف الأمصار.
Rasm al-Qur'an رسم القرآن
This is the science that discusses the letters as written in the Qur'an by the sahabah: Often we find additional letter or letters that have been left out, beside this we know that the Arabic letters have passed through a certain development, as in the early days of Islam letters such as ب, ت, ن, ي or س, ش or ف, ق and و were hard to distinguish without advance information due to the absence of the punctuation for example ي and ب inside a word (ـبـ, ـيـ) like أنجينا could hardly be distinguishd by a non-Arabic speaker as this word would look like 
. Nevertheless scholars consider this -in our (modern) understanding wrong- scripture as canonical, so even modern moshafs stick to it, as and I won't get tired of repeating it the Qur'an is the orally transmitted word of Allah not the scripture inside the moshaf:
Rather, the Qur'an is distinct verses [preserved] within the breasts of those who have been given knowledge. And none reject Our verses except the wrongdoers. (29:49)
(See also for example Why do words like "salah" and "riba" have an extra unpronounced letter waw in them?)
Now and this is the actual topic of your inquiry not the topic of qira'at so we have started our inquiry from the wrong perspective:
Hafs vs. Mushaff al-Imam and Kufa codex in (6:63)
In the following i will try to explain the meaning of the statement of al-Azami accoridng to major sources from tafsir and rasm books and to explain how the kufi recitations have taken it into account:
If you are referring to this form
- then you should know that أنجينا is -more likely like in the picture above showing the whole verse- the original codex of the Mushaf of al Kufa or as it's not that far from the scripture of the Mushaf al-Imam an equivalant scripture and the "pending" alif over a yaa' (without punctuation) which has been added in the "rasm" (note that I was not able to type a yaa' without punctuation to write it more properly, that's why I edited it via paint in the -first- small picture above, so when ever I write it with yaa' I'm referring to this picture), is an indication for the Version of riwayat Hafs انجــانا, as the original script wouldn't make sense in our modern Arabic scripture.
- While the same Verse in the Mushaf of Medina among others includes أَنْجَيْتَنَا.
- And according to the tasfir al manar the scripture in the mushaf al Imam -the copy 'Othman kept for himself- was (as you quoted) أنجينا, this is the only reference from a tafsir book to rasm al-Quran I know of:
(وهي مرسومة في المصحف الإمام هكذا ( أنجينا
But this differs clearly from the statement of al-Azami saying that that the difference is between actual Quran copies and original copies of the Kufi codex, which is taken from ad-Dani's book about rasm al Mushaf المقنع في رسم مصاحف الأمصار "al-Muqni'" (pages 44,48,52) and abu 'Ubaida's فضائل القرآن "fadail al Quran".
وفيها في مصاحف أهل الكوفة "لئن انجينا من هذه" بياء من غير تاء وفي سائر المصاحف "لئن انجيتنا" بالياء
والتاء
(This is a -typo- corrected Version from the linked pdf of ad-Dani's book.
Note
In the commentary on Abu Dawod's kitab al-Mashif كتاب المصاحف this statement is quoted while he clearly confirms the scripture as it is still known (for example in the Copy of King Fahd): This is what i found also in the scanned printed copy of al-Bahr al-Mohyt, and also in this scanned copy of al Muqni' (pages 97, 107 and 116). On the other hand the other readings from al-Kufa (Khallaf, al-Kassa-i and Hamza) are pronouncing the alif with Imalah which sounds near to a ya' (you could listen to them here).
Now if we take a look into the poem about rasm of Imam a-Shatibi called عقيلة أتراب القصائد في أسنى المقاصد we find
والكوفي نجيتنا في تائه اختصرا...
Which literally means that in the Kufi codex the ta' تاء has been recessed.
Which is also in ibn Ashir's تكملة مورد الظمآن في رسم القرآن
... وهنا قد حذف الكوفي تا انجيتنا... وشركاؤهم ليردوهم بيا...
Finally imam ibn Mihran ابن مهران -IMO- clears the issue in his book on the ten qira'at al-Mabsut المبسوط: (195 - 196) saying (My own translation take it carefully):
قرأ أبو جعفر ونافع وابن كثير وابن عامر وأبو عمرو ويعقوب {لئن أنجيتنا من هذه} [63] بالتاء.
Abu Ja'afr, Nafi', ibn Kathir, ibn 'Amir, abu 'Amr and Ya'qub recited {لئن أنجيتنا من هذه} (6:63) with a taa'.
وقرأ عاصم وحمزة وخلف {لئن أنجانا} بالألف من غير تاء، وعاصم يفتحه، والآخرون يكسرون"، ومقصوده بالكسر: الإمالة .
While 'Asim, Hamzah and Khalaf recited it {لئن أنجانا} with an alif (ا), but without a taa' (ت) and while 'Asim is reciting it with a clear fathah (َ ) sound (on the jeem "جَ") the others are reciting it with a kasrah (ِ ) sound (on the jeem "جِ"). And the meaning of the kasrah sound is (to create an) imalah.
So the yaa' (ي) was actually present in the moshaf of al-Kufa -as displayed in my small picture above- and it was interpreted by some of the reciters as an imalah. for whatever reason al-Kissa'i's -kufi scholar- interpretation was left out in this quote, but you may include him in the statement "the others" as he also applied the imalah!
And the meaning of al-Imalah can be found in:
والإمالة: تقريب الألف نحو الياء، والفتحة التي قبلها نحو الكسرة [ولذلك رسمت الكلمة بالياء]، انظر: الكتاب، لسيبويه: (2/ 310)، والشافية الكافية: (4/ 1970).
And al-Imalah means to make the sound of an alif come close to that of a yaa' by changing the sound of the fathah into a sound of a kasrah and that's why it was written (in the rasm of the moshaf) as a yaa' (Refer to Sibawayh al-Kitaab 2/310 and a-Shafiya al-Kafiyah) (source: islamqa #252582)
Conclusion
First of all the quote of al-Azmi is not about the science of qira'at, but about the science of rasm al-moshaf!
Secondly the statement is somewhat correct as the yaa' which was mentioned in the moshaf of al-Kufa is not present if we expect a yaa' as we know (with the punctuation), else the yaa' actually is indicated with a pending alif to be close or conform to the recitation of the people of al-Kufa. This yaa' in the rasm or codex of al-Qur'an was interpreted by three of the four known and accepted rway's from al-Kufa by the application of imalah on the jeem, while 'Asim recited it without any imalah (therefore neither Sho'aba nor Hafs have transmitted the imalah)!
Finally note that expecting any reading of the Qur'an to fully coincide with the rasm of al-Moshaf is a wrong expection as the rasm was kept due to the fact that many scholars conisdered it as canonical nevertheless they knew it is not going ahead with the correct recitation as all scholars agree that the Qur'an is and has to be transmitted orally not via a scripture nor based on it.
On this page (in Arabic) there are other differences between the accepted Qira'at and the Mushaf al Imam.
Here a Fatwa and also on islamqa about the rasm of mushaf. And i think reading Should the word دَسَّاهَا have a ي in it in سورة الشمس ayat number 10? could also be helpful!
And Allah knows best!
