The quote I responded to specifically mentioned slope. Illustrated by a standing on stairs analogy. That's vertical separation. Standing on a 45 deg slope you have a lot of vertical separation.
I grasp the concept, just don't like the name. It has nothing to do with vertically usually or standing on stairs.
It sounds to me like you still don't quite get it Tog. Vertical separation is not about the slope. Vertical separation is not observed from the view point of the horizon line, nor the snow surface. It is observed from the view point of the skier looking down at his feet. if he has long leg, short leg, then there is vertical separation. It has purely to do with making one leg shorter then the other one by flexing it, while not flexing the other one. Nothing more.
Long leg short leg....ie...vertical separation..can be achieved with or without lateral separation of the legs, which would be about spreading legs open vs keeping them closer together or neutral. I believe Rick was not referring to that kind of lateral separation though I think he was talking about lateral separation of the skis on the snow. That can happen by either using large vertical separation while still maintaining a narrow lateral leg separation, in concert with inclination. It can also happen by spreading the legs wider and no need for any inclination. It can also be a combo of the two.
The reason the term has meaning is because, like the herminator video provided, its often advantageous to maintain a relatively narrow lateral distance between the legs, in a bio mechanically neutral way. In that mode, we can see large vertical separation, which results in large lateral separation of the skis on the snow, but not large lateral separation between the legs. And there are other advantages of this mode in that shortening the inside leg helps to create angles, the very action of doing so contributes to the skiing. This act of shortening the inside leg, is the creation of vertical separation. Thus the term.
When I think of using the term "lateral separation", what comes to my mind is it must mean spreading the legs wider. There is a place for that too, when a wider stance is sometimes desirable during transition. I almost NEVER think about intentionally trying to widen the lateral separation of the skis on the snow. Let them be whatever they are going to be depending on slope, inclination, leg seperation and vertical seperation. The space between my legs and the vertical separation between my feet are more interesting things to think about as they directly effect balance and turn creation.