Good questions, Rusty!
You hear this one a lot these days. It's the concept of the boot shafts/shins moving (tipping) simultaneously and in parallel. It is the result of very active and disciplined inside leg activity, along with good boot setup. And the absence of "parallel shafts" (ie. an "a-frame" or sequential leg movements) is a reliable sign of an inactive/undisciplined inside leg, and/or a boot alignment/canting problem.
A rotary pushoff is a turn-initiating move involving "rotation" (usually some part of the upper body) and an extension/"push-off" from the "platform" created by strongly engaged edge(s). It is a "one-two" move--one part of the body starts turning while the skier remains planted on the "platform," then the skis turn after the "pushoff." It can be sequential (one foot at a time) or simultaneous.
When "sequential," the skier stands on the downhill ski while stemming/stepping the uphill ski, and usually also rotating some part of the upper body. Then the skier transfers weight to the uphill ski and "closes"--brings the inside ski parallel to the outside ski.
A "simultaneous" rotary pushoff is exemplified by the classic hop turns we see so often in the "extreme" movies. Hop turns do not necessarily involve rotary pushoff, but the typical hop turns where the skier throws his/her upper body around are clear examples.
In any case, rotary pushoffs affect balance in that they involve the body (center of mass) moving toward the OUTside of the turn. They cause a "glitch" in the smooth flow of the center of mass through the turn transition. In sequential rotary pushoffs, the weight transfer is part of the move. Rather than allowing the pressure to flow smoothly from one ski to the other as a RESULT of the g-forces of the turn, the skier actively throws the weight to the outside ski.
Because rotary pushoff falls in the broader category of "Rotation," it has the same effects on body alignment. Typically, the skier ends up "rotated"--the outside arm leads through the turn. This pulls the shoulders around, and the hips move out over the skis, creating edge-control problems.
Of course, many skiers initiate their turns with rotary pushoff, then COUNTER-rotate at some point to finish the turn facing downhill. This is a common movement pattern for strong and athletic self-taught skiers, or strong skiers who have not learned to steer their feet and legs independently of their upper bodies.
"Balancing v. Bracing"
This is another one I hear a lot, and frankly, I think everyone I've heard it from has a different definition in mind! To me, a skier who is "bracing" is NOT in balance--on either edge. They are usually supporting their weight on the inside ski while extending the outside leg rigidly out to create edge angle. But because that outside leg is stiff, and the skier is not balanced on its edge, it skids ungracefully across the snow. There is little, if any, control of turn shape. Because it keeps skidding away, the skier never becomes balanced on the outside ski, and avoids falling simply by virtue of the usually flat inside ski supporting his weight.
Again--good questions. There are probably no "right" answers to any of them. Good questions to stimulate some discussion though!
Hope you're enjoying the snow, Rusty!