The clarifications helped a great deal.
I see the waist affecting stability a couple of ways. In 'Cloud Hands' for example, the way I do them, with the waist facing the same way while stepping and turning the shoulders, allows each foot to be placed once with no adjustment, and allows the legs to be consistant with no torque on the knees. Another way I see it is in the smoothness of the coordination, ie. the leverage I apply with my upper body and arms doesn't "throw off" what my legs are doing, and vice versa. In these cases there's a certain consistancy that I attribute to the use of the waist.
You wrote that I mentioned that in the "ding shi," of 'White Crane' I have "shoulders right" and "hips left."
Actually I have 'shoulders left/hips right' at the 'final posture' of 'White Crane.'
You wrote, "In many movements, the amount my shoulders waist and hips turn to one side or another is certainly different. Perhaps, this is what you mean by the hips going one way and shoulders another."
You're on the right course with this, as this is part of what I mean. It's HOW that "amount my shoulders waist and hips turn to one side or another" is done, ie which muscles are used.
You mentioned, "As I move into the beginning of Brush Left Knee, my waist turns first, which allows my shoulders to turn more, but which leaves my hips pretty much behind." That's exactly it. Though your hips are indeed left "pretty much behind," that's where the leverage is, kind of like holding a bottle still with one hand and unscrewing the top with the other. The hand holding the bottle won't wander too far if you put no effort into it, but you can brace the bottle or twist that hand and greatly increase the energy applied to the bottle top. The picture of bracing the bottle fits the stability idea, and also points to those applications where the hips are not moving at all, but the muscles in the area are contributing torque.
Michael, Yes I do the Tung Family version. You are right about the "new" 'Single Whip' being old. There are videos that can clarify this. This 'Single Whip' Tung Kai Ying can be seen doing on his tape, and it is much the same as the single whip that his father Tung Hu ling, and his grandfather Tung Ying Chieh, can be seen doing on the "Dong Family History" tape. Tung Ying Chieh learned from Yang Chen Fu, so if I were to guess, I'd say that it is fairly likely that YCF did single whip this way at least some of the time.
Most of the applications that I've seen for 'Single Whip' are for dealing with two opponents. Michael notes two different energies and directions, perhaps seeing them applied to two different problems helps.
I should have titled this thread, "The waist is a terrible thing to mind."