If you tweek your terminology a bit here at epic more people would be able to relate to your valuable message that now gets a bit muddied due to un-necessary confusion.
I hear you TDK6, I'm not a technical writer and explaining advanced skiing technique through text is very difficult. I also realize that using video to represent intention or effort can also be misleading and confusing.
Take for instance when I describe the turn finish as closing the hips/extending the knees/closing the ankles. That is the "conscious" active movement/mechanics, but when we look at your very nice time lapse sequence, it is clear the skiers knees are actually "more flexed" at the turn finish. I believe you understand that this is due to "compression". The skiers intention/effort is actively extending/opening the knees, yet the knees are visibly "more flexed".
When I attempt to describe turn movements, I am always describing the skier's active intention of pressure , what the skier should be "thinking" about and actively executing.
I would bet that your feet are relative to your knees more back when your knee joint is extended than when its flexed
Starting form a flexed position, if I open/extend my knees, I will drive my feet forward in relationship to my knees.
Here's where I believe some of the confusion is coming from. I really like your drawing and will use it to explain, although you are going to have to look at it from a different viewpoint I believe. Remember, a lot of the flexion/extension movements are very subtle but are visible in line drawings that can negate the "compression illusion". Let's take a look and I'll try to better explain the message I'm trying to convey.
I'm going to separate the drawing into 2 different parts that I see the drawing representing.
IMO, the upper 2 skiers (A & B) in the graphic clearly represent the skier entering and at the turn finish.
The bottom 2 skiers (C & D) represent the skier at/through transition although the movements aren't as apparent and skier "C" should be on the right side of the image as it is earlier in the transition phase.
Skier "A" represents the skier just prior to the turn finish. Skier "B" represents the skier at the turn finish.
At the turn finish, skier "B"' has "driven" his feet forward by extending the knees into the finish and his feet are clearly closer to the vertical plane of his knees than skier "A" who is entering the turn finish. The skier's knee angle has opened from approx. 90 deg. to 100 deg. as the feet were actively driven forward. The skier has also closed his hips from approx. 80 deg. to 70 deg. Remember, a lot of these movements are very subtle, but combined with the forces of gravity pulling the skier downhill, they can generate quite a bit of force.
To me this is very clear. The skier "drives" the feet forward by extending the knees into the turn finish which results in the feet being closer more aligned to the knees, more forward in relation to the hips also. Notice how the feet and the center of the chest are still vertically aligned, this is very important which will enable the skier to not get thrown backseat into transition, that's why the hips need to close so the chest stays in relative alignment with the feet, they both move forward in sync. Does this make sense? You will see the reverse is similar when the skier opens the hips and retracts the feet through transition, the chest actually moves slightly backwards to stay vertically aligned with the feet that are being pulled back with flexion/retraction.
Ok, let's move on to the OP's topic, "bumps drive hips forward on leg extensions or flick heals back", that I can try to explain using the bottom 2 skiers (C & D) at/through transition.
Here's why I disagree with Skidudes (CSIA) approach and why I suggest "opening the hips/drive hips forward while flexing/retracting/flick heals back" and opening the ankles as the best technique to retain/regain ski shovel edge contact at/through transition.
How did skier "B" at the turn finish/pole plant get into the position represented in skier "C"?
Because skier "B" opened/extended the knees and closed the hips at the turn finish, the skier now has motion range to do the reverse/opposite during transition, the float phase. The skier opens the hips which projects them forwards in relationship to their center of chest and flexes/closes the knees and simultaneously opens/extends the ankles as they try to regain/retain ski shovel contact with the snow surface that is dropping away from them.
Here is another, for no better term, "illusion" that can occur relating to the movements at/through transition in natural terrain.
In the drawing skier "C" has opened/projected the hips down the fall line and clearly the skier's knees have extended to create the "long legs" that Skidude wants to generate. How the skier got to this position and where ski pressure is at this frame/instance is where we disagree.
Remember, my descriptions of movement sequences are what the skier should be consciously focusing on with active muscle tension.
Immediately after the turn finish, the skier wants to flex/close the knees and open/project the hips forward down the fall line and open the ankles in an effort to feel the ski shovel edges make contact with the snow. The more energy/speed the skier has and the pitch of the slope really are what determine the amount of muscle range the joints must utilize and the length of time they must exert the movements/tension until gravity pulls the skier back down to the snow and the ski shovel edges hook up and engage the new turn.
Opening/projecting the hips down the fall line are what generate the "long legs", not actively opening/extending the knees. This is the critical movement sequence/intention that is missed by so many skiers challenging natural terrain. It certainly looks like the mogul skier is actively and intentionally opening/extending there knees during the float phase of transition straight down the fall line, when they are actually using knee flexion tension to pull their feet beneath them and are opening their hips which rotates their femurs into the vertical plane. Basically the entire time the skier is weightlessly floating through transition, they are constantly exerting knee flexion tension and opening/extending their ankles until they feel their ski shovel edges make contact and engage in the snow. They are often evenly balanced gliding weightlessly down the fall line with the tails of their skis off the snow and their shovel edges beginning to engage through transition. The skier should be trying to increase shovel edge pressure through transition.
The skier will soon begin to feel more shovel edge pressure building up as their weight comes down and will then be able to start closing their hips, which projects their chest downhill in front of their hips, and start slowly extending their knees into the middle of the new turn.
I believe this is how a skier most effectively retains/regains ski shovel edge contact at/through transition in natural terrain and when making QCT's on the groomed for that matter.
The promoted approach, by Skiddude and most others IMO, to actively open/project the hips while simultaneously actively opening/extending the knees will put the skier in the backseat with their weight aft and result in a static pivot/skid/slip, here's why most skiers have trouble in natural terrain and can't get or stay forward.
If you actively flex into the turn finish, edge release and direction change must be gernerated by bouncing or slamming the terrain below because to much of the flex range of motion has already been used up which must result in a pivot for direction change. The skier can't retract to regain/retain shovel edge pressure because flexion has already occurred and the skier is forced to open/project the hips, open/extend the knees and often open the ankle in an effort to "reach" down the hill to regain/retain ski edge contact.
The timing is off and the movement sequence isn't in sync with solid turning mechanics.