Yes, she would mean Univeristy of Michigan, because we Spartans never say we went to Michigan. Goooo Greeen!
Princeton is the most prestigious on your list. Then Georgetown. Michigan is a great school, considered a public Ivy. Vanderbilt is a nice campus, and although in the south, you can still get a decent pastrami on rye at Noshville and the downtown area of Nashville is very pleasant, it still is southernish, but not deep south at all.
I would think about where you might want to live, because you will make some connections while in college, and also take advantage lf colege giving you the opportunity to experience something different. I grew up on the east coast, and went to school in Michigan, and I think it was great being with midwesterners, and learning more about another part of my country. Although, the schools you named will draw students from all over the country, so not matter what the student population should be somewhat diverse in that way.
Also, go visit the campuses. You might get a real feel for whether you feel at home at a specific school. If it is truly a campus setting, like a mini city all its own, or if it is more of a series of buildings within a larger city.
In terms of smaller vs larger schools, I loved going to a big school, and I did not expect it. I did not drink or soke or toke, I never thought I would fit in at a big school that had a party reputation, but I loved it, i loved that campus and dorm life was really big at my campus. Definitely live in the dorms the first year.