terminology – Are Product Designers UX Designers who also do Product Management? If not, what’s the diff between the 3?

I’m having to work with Product Managers for the 1st time and we seem to be in a push pull of who does what and who has the final say on this or that. Someone argued that it’s bc I’m a UX Designer rather than a Product Designer. Every time I read an article about the diff between UX and Product design I don’t quite get the difference…

However, if Product Designers take the responsibility of a Product Manager as well, then I understand the difference. Conducting user research, ideation workshops, designing as well as being responsible for the backlog (I don’t mean contributing, I mean responsible for making sure all the ACs are there etc), writing release notes… Having to attend all those meetings with different areas of the business I honestly don’t know how one would have the time to design and do all that. Only in this case I’d say yeah, I’m quite happy with the title of UX Designer!

Mind you: I work on a very complex B2B SaaS product, not a shoe shop… I think that might influence the answers…