So first of all, I was the one who originally changed the start of the Internship Arc from chapter 122 to chapter 115 last year in June. My reasoning behind this was that there is a clear cut between chapter 114 and 115 which indicates the start of a new arc. Plus, Overhaul, the main villain of the Internship Arc, has been introduced in chapter 115. On top of that the chapter is focussing on Twice, who ends up making quite a lot of appearances during the Internship Arc.
Recently someone changed it back with the reasoning that it explores the aftermath of the fight between All Might and AFO, as well as the Internships only being explained in chapter 122.
First of all, the Internships already get mentioned in 121. Second, this reasoning only works because the community chose the name Internship Arc, which was chosen back when someone chose chapter 122 as the first chapter of that arc. However, I see a similar case as the Vs. Hero Killer Arc here, which was formerly named Field Trip Arc. In that case, the word field trip didn't really describe what the arc was all about. It wasn't about the field trip itself, but mainly about the confrontation with Stain in Hosu. So someone decided to change the name to "Vs. Hero Killer Arc" which imo fits the arc better.
This is a really similar case. While the students are going on Internships, the main part of the Arc is more about saving Eri from the Shie Hassaikai and not the Internships themselves. So imo, there should be a more fitting name for that arc, like "Overhaul Arc" or "Shie Hassaikai Arc" or sth along those lines.
Renaming the arc also removes the issue of the Internships not being mentioned until chapter 121, because the Shie Hassaikai were already shown in chapter 115.
Also about the reasoning that chapter 115-121 explore the aftermath of All Might vs AFO, IMO the Internship Arc explores that aftermath way more than the Provisional License Exam Arc does. How? Overhaul and Shigaraki are basically competing for the spot that AFO left open. The Provisional License Exam Arc itself doesn't really focus on the aftermath as much. Actually apart from a few lines from Aizawa in chapter 98, it doesn't focus on the aftermath at all. The only chapters that do are chapter 115 to 121 which could just as well belong to the Internship Arc.
As some things with them definitely feel out of place with what's happening later in that arc, I suppose another option would be making chapter 115 to 121 a mini arc that doesn't belong to either. This would probably be the better option to make both sides happy.
As for the reasoning that S3 will most likely end with chapter 121, I agree, however there have been a lot of cases where anime ended mid arc before. Again I'd like to refer to the clear cut between chapter 114 and 115 which imo clearly means that a new Arc is starting. Or has anyone ever seen a clear cut like that right in the middle of an arc? To me this cut clearly states that Horikoshi intended to start a new arc. Why else the cut?
TL;DR There are two options imo:
1. Rename the Internship Arc to sth else more fitting and change its start to chapter 115.
2. (Probably the better solution to make both sides happy) Make chapter 115 to 121 a mini arc, which doesn't belong to either the Provisional License Exam or the Internship Arc.