Internship is just the first year of residency, basically. There are some specialties where you do an intern year that’s pretty much seperate from your residency, but nobody on the show is doing one of those specialties, so we’ll just ignore them.
The residency year begins July 1st, for everyone. The med students have their own timeline, depending on what year they are and whether their school does the rotations by calender month or by four-week blocks, but all the residents start on July 1st. Graduation is usually in late May, and since few people do their residency at the same place as they went to med school, June is usually taken up with moving house and getting used to a new city. Orientation is usually the last week of June, then they throw you in at the deep end around 7am July 1st.
I don’t remember the deal with Carter fussing about getting into surgery, but it was a long time ago. If you’re remembering correctly, it could be a couple of things. They might have been trying to keep the timeline of the show more with the timeline of the tv season, and been more concerned with keeping the show sort-of realtime than with being accurate about how residency works. It might also have been him fussing about getting to do an acting internship with the surgery department before having to do his residency applications.
An acting internship is when a fourth-year med student spends a rotation acting as an intern. They don’t have quite as much responsibility as a real intern, but more than they had when acting as a med student. UK required you to do at least two, one in medicine and one in surgery. If Carter was trying to get into a surgery residency, he would have wanted to do his AI in surgery as early in the school year as possible, so he could have a good grade in it to put on his residency applications. See, applications go in around October or so, so the programs you apply to really only see grades from September or earlier. Most med students try to arrange their fourth-year schedules so they can get in as many rotations related to what they want to do as possible before October. That comment might have just been TPTB letting Carter know he’d succeeded in getting an AI in surgery for September instead of later in the year.