Yes, both "thank you" and "thanks" should be accepted. The Hungarian "köszönöm" does also have a shortened version, "köszi" (or "kösz"), but it is not a one-to-one with "thanks". I am trying to find a pattern here... I think "köszi" is much more informal than "thanks". Definitely do NOT put it at the end of an email. Use it with friends and family only.
And, when addressing someone by name, the more formal version of one's name goes better with "köszönöm". To demonstrate with an English name:
Köszönöm, William - Thank you, William - perfect
Köszönöm, Bill - Thank you, Bill - just fine
Köszi, William - This one sounds weird
Köszi, Bill - Thanks, Bill - fine
So, yes, ther is some formality involved, on the Hungarian side.
Most common male names in order: László, István, József, János, Zoltán, Sándor, Gábor, Ferenc, Attila, Péter, Tamás. Among newborns: Bence, Máté, Levente, Dominik, Marcell, Noel, Dávid, Dániel, Milán, Áron, Zalán.