Here's pivoting per the CSIA manual:
Skill - Pivoting
Pivoting is the ability to utilize the legs and feet to change direction. Whether the turns are carved or skidded, the lower body leads the turning effort.
Pivoting and direction change
Turning the legs in the hip sockets is the most efficient way to place the skis across the direction of travel. Combined with the grip of the skis against the snow, this deflects the skier in their arc. The result of pivoting is separation between the upper and lower body.
Pivoting and eding
Upper/lower body separation enables skiers to balance on their edges through angulation. Turning the femur in the hip socket allows the hip joint to be more mobile laterally to the inside of the turn, thus helping to balance on the edges.
Pivoting and balance
Turning with the legs helps to change direction efficiently without disturbing balance. Turning the skis without hip or upper body rotation also helps create balance on the outside ski.
(CSIA Manual, 4.12 - 4.13)
Notably absent: anything saying you must twist the skis on a flat ski, or that pivoting only happens at the beginning of the turn. Hence why I think there's a difference in nomenclature between CSIA, PSIA and even CSCF.
If some instructors say that pivoting is twisting the skis, then yeah, we're going to be understanding our definitions differently. (That's one of the issues I suppose Canadian instructors face in a primarily US-based forum.) For that matter, I don't want all of my learners using the PSIA version of pivoting in every turn because it wouldn't be appropriate in high performance skiing. But take heart that it's just a label; the mechanics of good skiing are the same regardless of the labels you use.