I define the rotation as rotating the skis to a new pointing direction from that in which they were pointing before you rotated them.
Yes, but wouldn't that definition describe either passive rotary of the ski in an edge locked turn or active rotary of the edges in a pivoted or skidded turn? The term "rotary" in of itself only describes so much. The term really needs to be supported by more equipment and/or biomechanical context before it means anything more than how Webster describes the term. There really is no authoritative definition in ski terms for the word "rotary" by itself. Again, you could be talking about active/passive, the ski and not the edge, the edge against the snow, the edge and the foot, the foot and the knee, the ankle and hip socket , the hip socket and the waist - shoulders - neck - etc.
Ultimately, what the ski is doing is far more indicative of rotary than what the body is doing as it is so indicative of everything else. Nothing is truly clear until final ski output is described. In that light, rotary is for either active and edge associated for skidding, smearing, pivoting, stivoting, scarving, redirection and friction for speed control - or - for for passive, skeletal associated movements for pressuring the outside ski and running that pressure migrating from tip to tail.
When we look at a graph of a turning skier from the top down, imagine three points: the BoS, the CoM and the pivot point of the CoM over the Bos. You would notice that, when the skier turns (carving), the CoM rotates on a parameter outside of the central axis of the BoS in the direction where we want pressure over the ski in any particular point in the turn. As the CoM rotates around the axis of the BoS our weight/pressure is applied to the ski whereever the CoM goes, and "swinging" over the four basic quadrants of the skis: Left shovel, left tail, right shovel, right tail. It migrates to the left shovel then left tail, return to center with 50/50 pressure in transition and "squared" up neutral with all rotary, angulation and inclination , then right shovel then right tail and return to center and so on ...
However, when we rotate our feet and skis against the snow under our body for any type of the rotary edging listed above, the CoM can rotate directly over the BoS, or the BoS directly under the CoM, with the CoM, the BoS and the pivot point between the two are all lined up and are without the parameter "swing" to control pressure over the four quadrants of the skis. The more the ski is skidding vs carving the more open chained our system is and the more difficult it will be to make powerful movements. Conversely, the more both skis are edgelocked into the turn, the more closed chained our system is. The more the system is closed chained, the more we can make the CoM "swing' the parameter over the BoS with much more powerful rotary and therefore have more control over our movements. In other words, our movements are much more pronounced, effective and efficient over a closed chained double edge locked pair of skis and is why I find it much more intriguing.
A visual below to the left of the four quadrants of the transverse plane that the CoM will travel with the use of skeletal rotation and tipping over the central axis of the feet or BoS. Imagine the vector between the CoM and BoS creating an imaginary "joystick" control over pressuring the ski. That is the CoM BoS relationship management that we talk about. Note that the CoM need not travel far over this transverse plane when using rotary edging and is, as well, not as able to do so as much in an open chained unlocked or less locked ski. Note that the travel of the CoM over the transverse plane looks more like a half circle residing in the frontal transverse plane,