I never cared for a downward backfist myself, (shoulder/elbow down, fist traveling straight down.) I call that a "vertical backfist. "
I always liked what I call a "horizontal backfist." Shoulder turned over, elbow up, fist travels outwards and at a slight angle up. (Like what you called a "rising hammer fist.") Essentially the only difference would be the hammer would connect with the bottom of the fist, backfist would connect with the back of the knuckles, adding a snap of the wrist as well. Very similar techniques.