Assuming the movement is directional, I feel two things happening at the same time to commence movement: (1) My hip settling, and (2) My body weight moving into the intended direction.
I don't particularly feel the need to "push" off the standing foot, especially as the commencing action. Indeed the muscles must activate to cause the body to move, but the way I interpret it as a sensation is simply of rolling my weight forward, backward, or in whatever direction I'm heading. It's not until just before I have left a foot that I really feel the ankle muscles engage in a strong way.
Separation in knees? Well, yes, inasmuch as the action is connected to the settling of the hip. One knee pushes forward and across while the other pushes back. Just make sure that this is accompanied by commencing of your body weight moving. The only time you should settle a hip while standing in place is when the next step is in place.
I've never heard of the concept of "pushing the center into the ground", but it was probably stated to illustrate some particular point. When I settle the hips to one direction or the other, I do feel that the center of gravity also relaxes downward. But I don't push.
I hope this helps.
Sincerely,
Jonathan
