For me, a fair is a gathering to celebrate a kind of particular "event" or "product type". You may get amusement rides and attractions as part of the fair, but it's predominately about the county/tractors/rose queen/whatever. Stafford has a county "show" rather than a county fair; I don't know if this is common?
A fun fair is solely about the rides/stupid fired food/hook-a-duck. Though some fairs like the Nottingham Goose Fair have evolved into a fun fair from a traditional fair about selling geese (or whatever).
A fairground and an "Amusement Park" are pretty close to me. Generally if it's mostly temporary rides in a fixed location, then it's a fairground. If it's mostly permanent rides but they're kind of dumped in with little to no theming - it's an amusement park. Hence Coney Island is a fairground, and Kennywood is an Amusement Park (or Coney Island was when I visited). Though that's just for me, I'll generally just use the term "fair" for pretty much everything anyway :lol: