"Wir reden nicht wie Lehrer und Schüler, sondern wie Erwachsene."
The translation "We do not talk like" suggests something in our individual speaking styles: "I don't talk like a teacher, but like an adult. You don't talk like a student, but like an adult." I would've translated the sentence as "We speak not as teacher and student, but as adults," which comes closer to describing the interaction between the two people--e.g., "We do not speak as if I am a teacher and you are a student, but as if we are both adults." Which--if either--would be a more accurate translation?
At first sight, I would actually say your first suggestion is exactly what the German sentences expresses. Your second suggestion should be fine, too, although one would probably rather use 'unterhalten' instead of 'reden' here. 'reden' is the actual act of speaking, so speaking about a certain way to speak seems appropriate to me.
sich unterhalten for 'conversing with so.', that's what I would use here. There are other meanings for that word though. unterhalten as 'entertaining' or 'maintaining, keeping sth. running/going'. Unterhaltungsbranche roughly equals 'show business'. 'Wir unter halten uns' – We're having a conversation.