Ozan, there are two ways to go about this. You can seek to refine your current movement patterns. Or you can look at the big picture, and decide if your current movement pattern is the best choice for the kind of skiing that you want to do, and possibly make some very fundamental changes.
In reality, there's a whole range of potential styles for turn transitions. You can make the transition with a big extension. You can do it with a big retraction move (flexion). You can do it with a movement that's somewhere in between, incorporating some degree of either extension or flextion or neither with each leg, independently. It's a continuum. There's also the question of where in the development of a turn you become most extended, and where you're most flexed.
The points that you raised in self-critique seem minor to me. I concur with T-Square that there's a nice flow to your movement. So the movement pattern that you're displaying is working pretty well for you on the terrain and in the snow conditions you were skiing when you made the video clip. Maybe the $64 question is, how well does that movement pattern function in other situations? Steeper terrain, hard slick surfaces, heavy sticky cut up powder, moguls? In a race course? And in what conditions do you want to ski? What are your goals? Also, do you currently ski with the same movements under all conditions? There are limits to the conclusions that can be drawn from watching a single video clip. We are forced to assume that your style is the same under other conditions, and that could be incorrect.
What stands out most in your skiing is that you are using quite a large extension move to facilitate the release from the old turn. Also, it's pretty vertical, as opposed to projected outward toward the center of the new turn. The effect is close to the old up-hup-and-around from the days of straight skis, although maybe not as complete an unweighting. Your skis do not become edged, weighted, and (critically) bent until almost halfway into the new turn. Notice how there is a point in each turn at which you visibly settle onto the skis and exert force on them, and how this produces an abrupt spray of snow--watch your clip in slow motion. Observe how your center of mass doesn't trace smoothly linked S shapes down the mountain. Instead, from the point that you settle onto the skis, they kind of shoot you across the hill. Here the skis sometimes get locked into a path where they are no longer turning nearly as tightly as higher in the turn. So your linked turns are characterized by sharp, hooky, pivoty starts, connected by straighter sections.
I will say that in the second half of the run, which is a gentler slope, you look much different. Your turns are much carvier, you are getting much earlier edge engagement, and your turns are much more rounded. You might take this as an indication that your current style is less well suited to steeper slopes.
How are your current movement patterns, with pivoty turn initiations, going to treat you on steep, icy slopes? When you are in untracked snow that you sink into, but that is sticky or dense to the point that it doesn't easily give way laterally?
There is more than one school of thought on what is the optimal go-to transition style, on that flexion-extension spectrum. Some go in for a substantial, two-footed flexion move. Others favor what Mike King has described--flexion of only the old outside leg to release, and then allowing the new outside leg to extend, starting after the edge change. I'm sure there are other variations on this. They all work. Pick one style, and work hard on only that for a while.
Your poling needs work. Your pole touches are a small fraction of a second late, and they are too far forward--too much of a pole swing and too close to the body. These pole touches are inhibiting you from moving your center of mass across the skis, and tipping your feet to create strong, early edging. Frankly the best thing would probably be just to lose the poling entirely while you develop the footwork, then add it back in later.
I concur with the observations of Mike King and T-Square that you're letting your inside foot come too far forward. For the reasons already discussed, you want to minimize inside tip lead. Work on tucking back and tipping over the inside foot through the entire turn, from the point of transition to the end. This will be the result of a combination of dorsiflexion--lifting your toes to close the ankle joint--and bending the knee to allow the foot to come back. Try this standing still in a traverse position on a moderate slope. You still want to keep your inside hip bone forward and out, so that your pelvis is oriented substantially down the hill, so don't bring the inside foot back by moving your hips. If you do it as described, by dorsiflecting the ankle and bending the knee, you'll feel quite a stretch in certain muscles in the leg that's on the inside ski. That's a cue that you can use while skiing, to let yourself know if you're making the move. This will naturally press the tip of the inside ski into the snow, so yes, it results in a certain "load" on that ski.
Watch the video clip closely, and you'll see that there are occasions on which your inside foot slips forward just prior to transition. This is really going to mess up your turn initiation. You want to be in a position to be able to balance on the little toe edge of the old inside ski just BEFORE initiating the new turn. The focus that you say you've had, of crushing the tongue of the outside ski's boot, can put you so far forward that you compensate by pushing the inside foot forward for balance. Instead try just staying centered. As a generality, inside foot management is the appropriate focus--the outside foot will come along for the ride.
