It just means student, not a particular level. I was an exchange student in Hungary, and we were called cserediak
Because that's not used when you say what or who someone is (with a predicate adjective or noun), at least not in the present tense.
A lány magas. "The girl is tall." (predicate adjective: what?)
A lány diák. "The girl is a student." (predicate noun: what or who?)
A lány kint van. "The girl is outside." (adverb -- does not show "what" or "who" -- van is necessary)
what would the difference between saying "the girls are students" and "the students are girls" be in a sentence like this?
There is no accusative in a sentence like "The girls are students". That is true both in Hungarian and in several other (Indo-European) languages of Europe.
The verb "be" does not take the accusative, but rather the nominative.