I have never heard an American refer to "university." They say "college" just about exclusively. This is regardless of whether the place they got their bachelors degree was actually a university or a college (also, as others have mentioned, sometimes "universities" are made up onof smaller "colleges"). Literally the only times I've heard Americans say something like "when I was at University" is when they are talking to a foreigner. It's always "when I was in college" or sometimes, "when I was in school" (if it's clear from the context that they are talking about college).
Likewise, nobody would ever say "university" to mean any kind of post-bachelors degree. They'd say Med School, Business School, Grad School, etc.