With the given sentence, "Nem, az nem telefon," how do you know whether telefon is definite or indefinite?
Yeah is there a difference in Hungarian between "No that's not a telephone" and "No, that's not the telephone"?
It might be because adding 'egy' in this sentence changes the meaning? i.e. "That is not a (one) telephone (but rather two telephones)".
"Nincs" is the negation of "van". The positive sentence would be: "Igen, az telefon." (="Yes, it is a phone.") You can't say "nincs" in the negative sentence, because there is no "van" in the original sentence.
We use "nincs" when we talk about something that isn't there, for example, when you enter a store to phone someone and the reply is: "Nem, nincs telefon" then that would mean "No, there is no telefon". And speaking about the sentence you've mentioned in your comment, it's wrong, it doesn't mean anything.