I would say that rotation usually starts slightly before toe touch with the application of "prelaunch tht" that torques the handle of the bat.
The arms get this action started AND stimulate the same action in the legs, the primary action being synched external rotation of both the back arm and the front leg. This starts the body coiling with good timing.
The synched upper and lower body action is what lets the hands also be the site of control for how the hips will fire by controlling the coil of the body and direction of weight shift.
NEXT there is the "drop and tilt" where "THT at launch" consists of the more active contribution of the forearms and tilt of the shoulders to accelerate the bat and finish the coil before the shoulders connect to the torso and are driven by the untwisting/reversal of coil.
So "rotation starts" with hands and legs at the same time, followed by hips and then shoulders. At "initiation", Shoulders primarily "tilt"/resist first slaved to hands (not hips/torso) and then are turned by torso unloading/uncoiling.