Could the phrase also be "Nem tanár vagyok, hanem diák"? It's my understanding that Hungarian verbs already contain the subject information, or am I wrong?
Yes, it could. You are right.
Szia, fmk64! Köszönöm szípen! Is the "Én" there for the purpose of stressing the fact that "I am NOT a teacher"?
Yes, that's righ - it is there to stress the statement. I actually wanted to mention it in my previous comment.
In english you need to say I am not a teacher but a student. Obviously in hungarian you can omit the egy
Why is there no indefinite article on tanar and diak?
You can say it either way - the indefinite article is not obligatory, like, e.g., in German.
pupil is not accepted for diák
Should probably be - but from the context it can be assumed, that an adult university student was mistaken for a teacher.
Is rather important in that sentence ?
You need I am not a teacher in english, you cannot say I am not teacher
Why is hanem sometimes but and but rather. Seems insignificant
Good sentence to know if you have makeup on at school.