A couple of thoughts about Paul Lorenz' description:
--1:01...toppling the body across the skis. During the turn the ab oblique muscles are contracted on one side and the opposite back muscles are stretched from the angulation and counter. The release of these muscles helps "spring" the body across the skis as well as getting some pop from the skis' rebound. He's actually falling down hill from his skis and trusting his skis to catch up to him--which they always do. Upside down skiing--showing your ski bases to someone uphill from you when you edge early to big angles
--1:17...extension of the new outside leg. I feel it more as allowing the new outside ski to ski away from the body and the leg is gently extended to keep up with the ski. Never a push or thrust extension which might break the ski's grip on the snow.
--1:44...aligning my center of mass through my outside foot. i.e., as little weight as possible on the inside foot. A test for this is during the arc of the turn very quickly lift the tail of the inside ski off the snow, just a momentary tap, to confirm that the angulation is adequate.
--1:56...angulation from lifting the inside hip, not just curving the spine. The inside hip (& shoulder & arm) can be brought up and forward to combine angulation and counter.
--2:00...note that the inside foot is next to the outside knee. Legs very close together, inside foot pulled back close to that knee.
Getting the hip down to the snow isn't the goal. The goal is to create such edging angles such that the hip ends up down to the snow. Just trying to get the hip low can result in hip dumping where the skis' edging angles aren't maximized.
Paul vs. Reilly...it appears that Paul is making a deeper retraction which might result in quicker edge-to-edge. Hard to tell from videos on different slopes, different turn radius, etc.