I think the difference is in the usage of the terms weight shift and swiveling the hips. What I am talking about is this:from a basic stance - left foot flat on floor, rt heel up @ 2"- I want to throw a rt-lhook-rt-2 lhook-rt- lhook combination. To throw the rt my wt goes onto the left leg; the hook requires the wt go back to the rt leg; then back to the left...etc... You know all that. I am referring to swift shifts- if you begin with the rt heel up its easy, a mere matter of alternately having one foot then the other flat on the floor. Do you see? I can get wt behind each punch and it doesn't slow me down and as I describe it you can punch for longer intervals without getting winded because you are merely turning your body.
Swiveling the hips...Right-on the punches you are really looking to damage with you turn all the way through with the hips, like when swinging a bat. When throwing fast sequences the movement is the same as a dancer uses to get a lot of movement from their ...uh...rear. For a test of balance try taking two paint cans- gallon size- and stand on them and try and hit the mitts for a second. If you don't shift the wt from foot to foot,and if it isn't shifted in the proper order you'll fall off the cans. But if you are doing it correctly you can hit the mitts as you would normally. Its a great way to check your balance and to ingrain proper technique in that large muscle memory.
Finally, about which punch egts the big power turn. Watch Toney/Holyfield,or Toney/Hembrick. In both fights Toney throws the exact same series of punches again and again. Roughly it was rt hook body, right upper head, left hk body left upper head, rt upper head left hook body. He changes the tempo every time; first the hard punch is the 1st rt upper and the 1st left upper, then it will be the 2nd hook to the body and 3rd rightr upper. He kept working the same squence over and over and landing the hard punches every time because they were never in the same place twice.