I've always refered to them as ball caps too. However there's a type of ball caps I call "farmers hats".
The FH ball cap has a tall and usually stiff front, (often with plastic venting in the back) and might say CO-OP or WHO FARTED? on the front.
Regular ball caps to me are rounded and soft, fitting the top of one's head snugly. To be worn with the bill facing forward of course, to shade the face and block the sun from the eyes. (It is legal to briefly turn the bill to the rear if the bill interferes with shooting a scoped rifle of course, or while speeding across the lake to your favorite fishing hole.) Fishing your lucky, now soggy, hat from the lake really slows one down, and nobody likes to wear a wet ball cap.
Never heard them called "gimmes" tho.