I don't think there are, or ever have been 'rules' - or not since having a partner you aren't married to was considered 'living in sin'.
My daughter has been with her chap for years but they have no plans to marry. This may or may not change, but it's their business, not mine. If I refer to them to others I just say 'Ermintrude and Dougal', as I'm unlikely to talk about them to people who don't know who they are. If there is someone new in the conversation I would probably add 'my daughter' and leave it at that. Context is usually enough to get the meaning over.
My mum had a man in her life after my father died. That was trickier, as calling him her boyfriend when they were in their 80s seemed inappropriate, but they didn't live together (and I have no wish to know anything about any sex life they may have had!) so 'partner' seemed confusing, and he wasn't my stepfather in any sensible sense, as I was an adult when my father died. Some sort of context was sometimes necessary though. I think I just burbled something about how he was my mum's significant other with a vague explanation. Nobody cares, really though.