Discussion Forum—A Way with Words, a fun radio show and podcast about language

Yeah, I'd go with chin-up bar, since that's what they're mostly used for. Not like you couldn't hang from your legs and do "scrunches" if you wanted to. I think they might be more well-know by brand names aot generically, for one example, see:

By the way, there's a difference between chin-ups and pull-ups. For the former, your palms face you. For the latter (which are more difficult for most people) your palms face away.

Maybe the best generic term would be exercise bar?