Actually the area around NYU is the perfect college town (Greenwich Village). It has the cute shops, every imaginable restaurant and bar and a park. You can find everything a college student would want. I personally prefer real college campuses and cozier towns.
As a life long NYer, I have never ever heard of someone say they live in NYC unless they live in one of the 5 boroughs, so that surprises me that someone from Westchester said they live in NYC
Not to fellow NYers, but to others. When someone from Washington DC talks to a non-Washingtonian, they say “I’m from DC.” When talking to a fellow Washingtonian, it’s Falls Church or Rockville or whatever. I don’t really hear Londoners and Parisians specifying the location on first reference, unless it turns out the other person actually knows the city. That was my point. Golly gee, when I applied to Sarah Lawrence back in the day, I naturally perceived it as being in NYC. Maybe if I’d decided to attend and ended up becoming a New Yorker I’d have have learned to insist it wasn’t 
@momonalaptop I’m not trying to argue here but like @citymama9, I also lived most of my life in NYC and never heard Bronxville referred to as part of the city. It’s in Westchester. NYC is Manhattan, Brooklyn, Queens, Staten Island and the Bronx, period. From Bronxville, you cannot take a subway into midtown. There might be some kind of suburban express bus and I’m sure some branch of Metro North stops there, but that’s clearly suburban transport. Getting into the city for Sarah Lawrence kids is easier than, say, from Vassar (where I went), but it’s still “going into the city”.
PS: IMO yes, people consider Cleveland Heights to be Cleveland. People in Lakewood and Shaker and such generally say they are too even though their kids don’t attend Cleveland city schools and they have their own police, etc. Obviously if they are speaking to someone from Cleveland they name their suburb.