I think that all of these concepts are important and useful, though they may apply differently in people of various backgrounds and and running speeds. For example, when I was trying to drive through the hips, I managed to not lift the feet and was getting more impact. To correct, I conceptualized picking up the knees, and the feet followed. Though, some people probably overdo the idea of picking up the knees or lifting the feet.
I agree that at certain speeds, specific aspects of form may seem more natural and just fall into place. I suspect that at other speeds one has to make a more conscious effort. I think that there is a substantial difference between running and jogging (though I can't really pinpoint the speed which separates the two). The same would apply to the other great running mammal, the horse, whose gait is substantially different when walking, trotting, cantering, and galloping. (Though, I suppose that horses really "just run"!)