On the meaning of the word Iman in Arabic
The word Imam إمام (singular of 'A'immah أئمة) in Arabic is a noun that refers to
- a leader, be it a religious leader or a political one,
- be it a leader for a simple act or a general leader.
- as leading also means giving guidance, a guide or anything (or anybody) that includes or is related to guidance could be referred to as Imam.
- leading also may mean correcting or repairing things so anything that corrects may be referred to as Imam.
- Also we refer to a reference or highest instance as Imam as it is giving the directions and orders.
The corresponding verb is the verb amma أَمَّ.
On the occurrence of the word Imam in the Qur'an and the related interpretations
This word appears in the Qur'an in three verses in its singular form :
So We took retribution from them, and indeed, both [cities] are on a clear highway. (15:79)
فَانْتَقَمْنَا مِنْهُمْ وَإِنَّهُمَا لَبِإِمَامٍ مُبِينٍ
Here most of the scholars interpreted "Imam" as a clear (guiding) path or highway.
[Mention, O Muhammad], the Day We will call forth every people with their record [of deeds]. Then whoever is given his record in his right hand - those will read their records, and injustice will not be done to them, [even] as much as a thread [inside the date seed (17:71)
يَوْمَ نَدْعُو كُلَّ أُنَاسٍ بِإِمَامِهِمْ
On this verse we find many interpretations some say that Imam here reefer's to record or book and take (36:12: "and all things We have enumerated in a clear register.") and (45:28) as an evidence for that this was the choice of ibn Kathir for example. While Qataadh and Mujahid said it refers to their Messenger or Prophet and take the verses (4:41), (10:47), (16:84) and (39:69) as evidences to support their claim.
And some scholars said Imam refers here to the book or scripture that was revealed to them and among those who chose this was ibn Jarir at-Tabari.
And some scholars said: this means that each of the people would be called forth with their Imams: this means for the believers their prophets() and for the disbelievers their tyrants and leaders they were following as Allah says: "And We made them leaders inviting to the Fire" (28:41)
My main source for the above is the tafsir of Sheikh Muhammad al-Ameen a-Shinqiti محمد الأمين الشنقيطي called Adwa' al-Bayan fi Eedahi al-Qur'an bil Qur'an أضواء البيان في إيضاح القرآن بالقرآن, which is a tafsir which tries to interpret Qur'an by Qur'an. Who considered the later opinion as the closest to truth.
So when Dr. Mustafa Khattab points at three possible meanings he is just pointing at the three (first) given interpretations. Note that guidance or leadership by a Prophet is not much different from that of a revealed book, as it is the duty of prophets to transmit the message (the revelation). And the interpretation that Imam here is equal to record, even if it sounds a bit far fetched, as their record is in first place a result of their deeds, but these deeds have been committed while or after they have got a guidance be it a good one or bad one. So IMO all these views can be brought together as on one hand every people had guidance by Allah trough his revelations and prophets or and on the other hand "misguidance" by the leaders of disbelief and based on their own strength of faith they have committed good or bad deeds which have been recorded and these deeds may later lead them either to jannah (paradise) or jahannam (hell).
... and all things We have enumerated in a clear register.(36:12)
وَكُلَّ شَيْءٍ أَحْصَيْنَاهُ فِي إِمَامٍ مُبِينٍ
Here most of the scholars say Imam refers to Umm al-Kitab the clear register. Which is a guiding or leading book as any other one is wrong!
The plural form 'A'immah أئمة is quoted in five verses in the Qur'an and always refers to leaders of both kinds good or guiding ones or bad or misguiding ones (these verses are: (9:12), (21:73), (28:5), (28:41) and (32:24)).