It would have varied between inns, you know. But if it's called "Saloon Door", I'd expect it to be used for Saloons, myself. I'm not speaking from experience, but when I think of a wild west in from the days of say, the Gold Rush, I imagine them to not have any doors at all, actually. Just a door way to get in, but no moving door. But like I said, it varies. When I look up "1800's Inn", I get some results that have doors, and some that don't. But those aren't all necessarily "Wild West" inns. I think that if you just use a saloon door, people will get the idea.
