The funny thing is that, contrary to how they act in their roles as Captain, Kirk was a studious nerd and a bit of teacher’s pet at the academy while Picard was a hard-partying drunk who not only participated in, but started bar fights.
Kirk would do his best to defend you in a bar fight and then would punish you after the fact, according to Star Fleet rules.
Picard would try to stop the bar fight from happening to begin with, would break it up if it escalated, but probably wouldn’t defend you specifically unless you had a good reason for being in the fight. He would only punish you if you were in the wrong and then it would probably be something more creative, more immediately punishing, and less impactful (career-wise), then Star Fleet’s regulations prescribe.