I think that cap and hat are often the same as in baseball cap or hat. I usually look at any head cover as being a HAT and in some cases is also a cap. If you are wearing a Bowler it is always a hat. Same would go for a TOP or Derby Hat. I think more confusing would be the words pants and trousers. It depends on what the purpose is for their use. Like you would never refer to blue jeans as trousers, or you would not refer to the bottom of a TUX as pants, but of course you could and
that would be ok