Are you confusing zenész "musician" (zene-ész: music-ist) and zeneszerző "composer" (zene-szerző: music-author) ?
This is not an exclusive question for this problem, but can anyone tell me which tenses write the verb "van" and which not? I just don´t know how to distinguish these situations.
Do you really mean which 'tenses', or do you mean which 'persons'? So far in DL we have only studied present tense. In the present tense, the first and second persons take forms of 'to be' when stating what something 'is', but the third person does not use 'van' in similar constructions.