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)

• A lányok diákok
• A diákok lányok

Pretty much as one would expect, given that you have "the" in front of "girls" in one of your sentences and in front of "students" in the other, and the predicate (are students, are girls) comes second and has no article.

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.