Sorry, I didn't mean the move tool I guess. What is mean is called "manipulator."
Specifically:
How do I set the pivot of a layer, without moving the sculpt in that layer, by setting the transform? How do I move the sculpt in a layer, without moving its pivot, by gripping it? How do I move the layer, moving the sculpt and pivot by setting the transform? How do I set the transform of the mirror plane? How I do grip and move the whole sculpt, including selected and invisible layers, pivots moving with sculpt? Can you undo after hitting the 'center' command? It doesn't do what I expected. It acts more as a 'center object to world' rather than 'center pivot to world.'
I'm sorry, but that video isn't very helpful. Without knowing the exact button presses involved, I can't replicate it. Also, the video makes it unclear if it is working on the entire sculpt, the pivot of a layer, the layer within its pivot, etc.
Also, the manipulator menu I get seems to be missing buttons from the one in the video.