Replying to this quite late but I do find it interesting none the less.
My father teaches at university and he used to give me a hard time about calling university "skole" as some people say you shouldn't. Skole should refer strictly to "folkeskolen" and "gymnasium". The same goes for "lektier", here "forberedelse" should be used at university, some would argue. And "timer" isn't used at "university", rather "undervisning" is. Lastly, the teacher at university wouldn't be a "lærer" but "underviser/professor/lektor" or w/e the title is.
But these are just the rules, in general I'd say that about 50 % use the "incorrect" terms and some people will find it snubbish if you correct it whilst others find it terribly embarrassing to use the simple wording.
Maybe it's a generation thing - I'm from 58 - but I wholly hold with the distinctions outlined by Rasmus, and would consider it an Anglicism or an Americanism, rather, if I heard university refered to as "skole"
further, you become "student" when you graduate from highschool - "tager studentereksamen" and then you may "studere" or "læse" at "universitetet" or "handelshøjskolen" or some other "højere læreanstalt"
and by the way, your teacher at "gymnasiet" was not a "professor" but a "gymnasielærer"