Log in/on log out/off are just historical hangovers from a computing world that existed some 40-45 years ago. The terminology was being invented then to match the technology. It may have been even earlier. There is no correct form, only historical variants.
It's a bit like the differences in UK and US English spelling. I try to use UK-EN spelling and I end up using US spelling...
But I would never, never ever log on.... only in. That is what we did that far back.... Whatever terminology is in use for a particular service just accept/know what it is and use it. Do not tilt at windmills! Let me do that.