I am curious what conclusions you've drawn from your own analysis, and what you were concsiously trying to do in these clips. From your description that it is "arc to arc" in the first two clips, I assume that this is what you were trying to do.
And you succeeded, in many respects. In those two clips, you ride the sidecut of your skis throughout each turn, cleanly, from start to finish. You move very quickly from one set of edges to the other--something I suspect that you intended. But why were you trying to ride "arc to arc" in the first place? In my opinion, "early edge engagement" and "arc-to-arc carving" have become way over-emphasized these days, often to the detriment of good skiing.
What it's done to you in these clips is make you very static--what's often called "park and ride"--with little control of your turn shape. You have two positions--a "right arc position" and a "left arc position." You move very quickly from one position to the other in the transition, then stay in that position pretty much motionless throughout the turn.
The result is that your skis take you for a ride on a path that they--not you--choose, like a train on a track. And that track is not consistent--it varies in radius (tightens) from start to finish, as the forces and the hill angle (relative to your skis) build throughout the turn.
You also tend to get to your new inside edges so quickly, before you've moved sufficiently inside the new turn to balance on seriously carving skis--forcing you to actually prevent
them from carving as well as they might. This latter is another reason why the beginnings of your turns tend to be rather straight, despite your skis being on edge.
My suggestion is that--other than when practicing specific drills and exercises--you stress the "early edge" and "arc-to-arc" thing a bit less. Develop patience through the transition--"float" and guide
your skis to their new edges. Before you try to engage those new edges, wait for two things: forces pulling you "out" of the turn (combination of gravity and centrifugal) and sufficient inclination (leaning into the turn) to balance against those forces, as your carving skis drive you through the turn.
So...as you exit the previous turn by releasing your edges, you will become "light." Savor this "float phase"--it's one of the most important (and enjoyable) parts of turns. It's a moment when you can relax, as you literally "fall" into the new turn, a time when all that matters is that you prepare for the forces that will build quickly once your edges re-engage.
In the "old days"--pre-deep sidecut skis--I used to recommend waiting all the way until you reached the fall line (straight downhill), or nearly so, before trying to really engage, pressure your skis, and carve the finish of the turn. With today's skis that bend and carve with much less pressure and edge angle, you can often carve earlier than that, but the principle is the same. Wait. Float. Be patient. Incline into the new turn, create new edge angles, but be patient before trying to get them to really bend and carve. Guide your skis during this phase, as your body moves inside the turn--no pivoting or pushing the tails sideways. Once you "feel the force," then go with it--engage and let the g-forces build and bend your skis.
These thoughts will require continuous, smooth, sustained motion, in contrast with the sudden stop-start moves of your video clips. It will require much practice, and a lot of "touch." Every turn is different, and the forces you deal with change constantly--due to pitch, speed, turn radius, and snow conditions. Managing these constant changes requires constant movement, but again, these movements respond to variables that you cannot always control, so they rely on "touch" at least as much as on technical understanding. Timing and intensity will vary with every turn!
Here's a very small animated clip of the movements I'm describing. (Sorry--I'm limited in the image size by my hosting service--this is as big as I can make it. Animations take space!)
Notice the transitions, which begin way back in the previous turn as the edge angles and body inclination begin to decrease, and continue seamlessly through the actual moment of edge release and through the "float phase" of the new turn. It's one unbroken movement. By the time the edges re-engage, I'm way
inside the new turn, where I'll be in balance as they start to carve. In the float phase, I'm guiding my skis as I tip them and incline my body into the turn, but I'm not in a hurry to get "heavy" on them or to carve "arc-to-arc."
In spite of this patience, the tracks will look "arc-to-arc." I think that's one of the reasons why many people--including many instructors--over-emphasize (in my opinion) the "early carve" thing. Relax! Be patient! Spend less
time on your edges and you'll spend a lot more quality
time on them.