Consulting the OED, "inn" comes from the Old English "inn", which means any sort of "habitation". It currently refers to a commercial establishment where one might lodge. It has been used in the modern sense since AD1400.
"Tavern" comes from the Latin "taberna"--a shed made of boards, a booth, a shop. Around AD1286, it referred to a place that sold wine (but had no lodgings). The term "public house" didn't appear until AD1669. "Alehouse" is at least as old as AD1000.
So, if you want a "high feudal" flavor or ealier, "tavern" should be restricted to upper-class establishments that serve wine. "Alehouse" to the common establishments, and "inn" possibly used for any place for lodging.
"Bar" is from the 1500s.