I never think of extension as being the method to transfer weight early to the new ski. However, after edge change, when the skis are on their new edges, as you are moving your body even more into the turn, pumping your legs, extending and reaching, you do add more pressure to the skis, both of them with that extension move.
So, before edge change, the inside leg extension is mainly the knee joint extending and not so much the ankle joint. The extension is facilitating moving the CoM into the new turn, towards the cross over point at transition/edge change. This extension is not transferring the weight early, but just beginning to move the CoM across and forward.
What is causing the weight to transfer are your own muscles - gradually balancing more against the new outside leg while it is still the inside leg. I try to explain it this way.
Stand up. Stand on 2 feet equally. Now slowly lift one foot. As you slowly lift one foot, more weight gets transferred to the other foot. Now, try this. Stand up tall, weight on 2 feet. Now flex slowly and while flexing start to weight one foot a bit more and then extend again. Now start to play with the timing until just at the bottom of your flexing is when you start to weight one foot more. This isn't exactly like skiing at all, but it gives you the idea of how an early weight transfer works. It's a balance thing more than anything else - how you are distributing the weight from your hips to your feet. This is why the hips are very important to be able to do any early weight transfer. If you counter or rotate your hips, you cannot as easily choose how to distribute the weight between the feet.
In skiing the exact timing of beginning the early weight transfer and the extending the inside leg varies with the turn dynamics - the shape, size, speed, snow etc. In a very dynamic turn, with much turn force, you need a strong, extended outside leg to hold the turn - a flexed leg is too weak. So, you need to really manage this turn force after the apex by flexing and absorbing enough of it so you can hold the edge as you guide the skis finishing the turn and also start to move to the new ski early: i.e. shortly after the apex.
So, does this clarify more the cause/effect issue between transferring the weight and the extension of the inside leg?
No it does not clarify because this is not what I am seeing. I see the new outside leg reaching full extension at transition/flat skis. I've personally found that opening the ankle joint as part of new turn initiation (i.e. equivalent to starting the movement at 16 seconds) enables some wonderful lturns. I can't do your exercise. In order to slowly lift one foot I have to make other simultaneous tipping movements to compensate. I can make movements with my ankles/knees, hips or head/shoulders. I'd really like to know how to distribute weight from my hips to my feet.
The point about the extension just moving the COM into the new turn is exactly the crux of the "trick" question. I believe that the old inside leg/new outside leg extension can be either purely a result of the hips pulling the leg down the hill (0 weight transfer), purely the result of weight transfer (the equivalent of taking a lateral step) or a BLEND of those 2 scenarios. The theory would be that to the extent the blend is biased towards the latter, you would have earlier weight transfer. Why was it common for racers to lift their new inside ski?