I always thought the difference is with a normal shove-it the board pivots on the back trucks, whereas with a pop shove-it all wheels are off the ground for the rotation. They seem to be depicting only the latter in the animations.
Popping, i.e. hitting end of the board on the ground, to get height is the difference.
A pop shuvit will be higher while a non-popped shuvit may barely leave the ground. Popping can actually make it easier to keep the axis of rotation closer to the center of the board.
I always did hard flips as varial kickflips, but with fs-shuv instead of bs-shuv.