You’re playing at being the mature, responsible person, telling somebody who is ill-behaved that their behavior is problematic. But you’re not actually being a mature, responsible person here, as evidenced by the fact that you chose to insert a parting shot in your “I don’t want to argue about this anymore.”
Logical fallacy: ad hominem tu quoque.
Do point out what argument I claim to invalidate there, if you would.
Or, more pithily: Fallacy fallacy.