> If you say "I am at school", it means you are a student, you attend school. If you say "I am at THE school", it means you are physically at the school building, without specifying whether you are a student. Both of those options are different from saying "at a school." See also "prison", "work", "bed". Similar distinction (in UK English, not so much American English) with "in hospital" vs "in the hospital."

Your description is incomplete. I agree that "I'm at school" carries at least a strong implication that you are a student at that school, but it also carries the denotation that the school is where you are currently physically located. If you want to specify that you are currently a student without claiming that you're currently on the premises, you would need to say "I'm in school", though that particular expression (a) doesn't specify which school you attend, and (b) can't be altered to provide that information (which is to say, you can't replace "school" with the name of a school). To get there, you need to change the verb; you can say "I attend Florida State", at which point the stative inflection on attend makes it clear that you aren't discussing your current physical location.

The hospital example is pretty much the same. Being "in the hospital" is analogous to being "in school", a fact about your current status rather than your current location. For your location you'd want to use "at", which doesn't have the same weird thing going on in "at the hospital" that it does in "at school".