Autonomic reactions do not produce walking, it is through active experimentation that we learn to balance on our feet and move in a specific direction. Be that walking forward, backwards, or even sideways. The only real difference on skis is again the bilateral quality of our ski movements. Hopping verses striding is the easiest way to describe this difference but even that get muddled when we start to talk about counter balancing efforts during a high performance turn. An idea echoed in the Whistler video that some may over look is the idea of cross lateral flexing and the movement of the inside shoulder that draws it toward the outside leg and how it's not always enough to flex the leg and draw the outside foot to the body. It's an idea Bobby Murphy and Eric Lipton mentioned to our training staff a few years ago. My point is CSIA and PSIA top pros have been talking about this sort of stuff for quite some time and it fits very nicely into the concept of the entire body participating in our balancing activities. That doesn't mean every ski move uses either exclusively though, a sliding scale where cross lateral movement is at one end of the spectrum and bilateral movement is at the other end might help explain the difference between the two but it needs to be understood that our ski movements usually include a combination of both and that balance / counter balancing moves involve both. For example; watch a WC racer get out of balance and it is likely you will see them throw an arm out to regain balance. Even though they are using bilateral leg steering move to steer their skis.
BTW, that phrase about the entire body participating appears at the beginning of every task descriptor used in our Certification training and exams. It also appears in the published movement disqualifiers used by the examiner and training staffs. It's one of the common threads and regardless of how we dissect a turn and discuss it's component parts, there is always an overarching concept of balance being an outcome created by the actions of the entire body and we are always operating in all three dimensions.
More specific to the comments about traction and how it relates to edging, skate across level ground, side step up a hill, or turn out of the fall line. Traction allows that to happen because the feet find something to push against and the surface pushes back. Without that all we could do is slide down the fall line, like we do on an inner tube. Stay open to that wider view of things and a lot of the misconceptions about skiing will disappear.
Balance is an outcome and regardless of how we focus on a particular part of that task it really doesn't change. We are keeping our butts off the snow by moving both our core and our feet. To what extent depends on the terrain but changing the DIRT doesn't change the task. Be it on a sidewalk, a bunny slope or a steep expert run.