Rush down is not a term that apply to a character, it is infact a playstyle of the player them self. So saying a character is a rushdown type is not right. Saying that character has strong factors to maximize the rushdown playstyle of a person is more accurate.
The term rushdown is used when the attacker continuously attack the defender. And this term has been used in other types of game not only in fighting game.
The reason people confuse these terms is because one term has elements of others. When you are pressuring, you are using: mix-up, block-string, frametrap... But you can't say pressure is mix-up. It just has the factor that the term mix-up define.
I think frame trap is just a more specific definition of the term bait. In a siuation where the attacker is pressuring the defender, the attacker purposely make a hole in his block-string, baiting the defender to counter them, then use a more frame advantage move to neutralize that counter.
It just specificly say that the attacker use the "frame advantage move". When Slayer make a hole in his block string, then use dandy step and follow up with pile bunker to counter the defender. You can't say he was using frame trap. It is more apropriate to use the term bait here.