WOW!!! That is a loaded, but fun, question to say the least. The English language has a million different names for the same thing.

I always call something a cap like a baseball cap or a ball cap. There is a visor attached to all of them.
A hat is anything else. Pillbox hat, cowboy hat, top hat, etc. This is just the way I see it and I'm sure there are many other ways to see it.