All the way up until that Voyager episode where Paris turns into a fishman, "transwarp" basically meant "Going Really Really Fast", so fast you were everywhere at once. Theoretically, you could make ANYTHING go to transwarp if you glued a fat enough engine to its ass.
Err no, right from the moment it was introduced back in TNG it involved them going elsewhere. Notrmal warp moves you through space time using a subspace feild. Transwarp actually drops you into subspace.