It’s “public”. But that would be the same as filming you in your own house. If it’s a friend you invited over, they could record you and it’s on you to indicate your opposition and kick them out/trespass them should they refuse to comply.

Now in the private bar, the other patrons are allowed to be there and there’s no law prohibiting them from recording (excepting places like a bathroom of course). If the bar tells them not to record, they can comply or be asked to leave. If three bar doesn’t tell them to leave, it’s on you to leave. Consider if a nazi walked into the bar. They have the right to be a nazi and go to bars. Bars have the right to refuse or provide service to whomever (so long as it doesn’t target a protected class). You have no more right to be at the bar than the nazi or person filming (absent some other condition like the bar along them to leave).

Tl:Dr - it’s not public in the legal sense. However civil law takes over.