Hi Adam--I know this wasn't your first post, but welcome to EpicSki!
You're basically right when you say that you could turn more with "the bottom part of [your] body and less with the top." But just keeping your upper body facing down the fall line is probably not going to accomplish it. Most of your turn comes from what is classically called "rotation," meaning you turn your upper body first, then pull your skis around second in a 1-2 motion. In your case, the "rotation" comes mostly from your hips and somewhat from your shoulders. (It looks like your effort to keep your head and shoulders facing down the hill is keeping your upper body movements mostly lower in your abdomen and hips, but the movements are still there. Notice in your earlier video that the rotation involved more arm movement.) There's a little bit of an "up" move as you rotate your hips, too, which lightens your skis and makes it easier to throw them around. You are not too far back, at all. In fact, you are usually levered forward a bit, also making it easier to twist those ski tails sideways. All in all, these movements create a very skidded turn. And as you've noticed, it is virtually impossible to make short, quick turns this way, because once you get your whole body turning (spinning), you have to then make it stop before you can throw it around the other way. Each new turn requires that you first stop the skidding of the previous turn--which is, as you've discovered, especially hard to do on hard, icy snow.
The solution is probably not as simple as thinking of turning your torso less. At the very least, it will require some brand new movements of your legs that I don't see happening in the video sequence, in which your legs rotate independently of each other in your hip sockets, while your pelvis (and everything above it) remains stable. That is the opposite of what you're doing in the video--turning your pelvis (hips) and upper body first to throw your skis around.
If you stand up where you are and just try turning one leg left and right, without moving anything else, that is the movement I'm describing. You can do it with either leg, or with both at the same time, and it can be powerful or subtle, quick or slow, and very precise. When you can do this, you will eliminate the need to turn your skis with your upper body, because you'll have a new way to do it. In other words, rather than thinking of "less upper body movement," think of developing more leg activity--which will eliminate a main reason you're turning with your upper body in the first place. (It is easy to describe, but unfortunately, it will probably require more than just this description to learn to do it on skis--I strongly recommend a lesson with a good instructor.)
Perhaps more important and more fundamental, you may need to rethink your turn from the ground up--right down to why you're making it in the first place. It truly looks to me like all the movements you're making in that sequence are intended to get the skis skidding sideways, very likely because you're looking for speed control, looking for those skidding skis to scrub off some speed. That's a pretty common thing, but it clearly isn't compatible with a goal of carving cleaner turns with less skidding. You didn't say that was something you wanted, but I'll bet it is. It will certainly help you ski those shorter turns on the hardpack without having to "bail."
Great turns hold the line, with minimal skidding and maximum gliding. When you skid less, you will have much more control of your line, which allows you to shape turns in a way that allows the mountain to control your speed for you, without you needing to scrub it off with skidding skis. Such turns demand subtlety to guide your skis precisely, rather than throwing them around with big movements of your upper body. You'll want to develop more refined and active edging (tipping) movements to get your skis to hold better and to help carve the turns. You'll need to let your skis glide and actually gain speed as you start each new turn, then keep guiding them around, across, and sometimes even back up the hill a little, so that gravity slows you down, not skidding.
While you are not too far back in the video, that is given the movements you're making. You don't want to be more forward when the brakes are on! As you learn to glide and carve more, you will need to move more forward with your skis. But don't put the cart before the horse.
Like I said, these changes involve a big, fundamental shift in thinking--a change in why you turn, not just how. Great turns are made to control direction, with speed only indirectly managed by skiing a "slow enough line."
These movements will be more like skating, gliding, slicing the snow, rather than making "hockey stops." If you are a skater (inline or ice), you have a big head start.
If this makes sense and sounds right to you, go play around with it, then come back with some specific questions. We may be able to help you here in the forums. Preferably, like I said, sign up for a lesson with a good instructor. That will be a lot easier!
Good luck, and keep us posted!