Don't worry about carving quite yet. It's an advanced edging technique that requires a broad basket of foundation skills to do safely and well. Very few actually know how to do it,,, even of those who say they do, For the time being, your focus should be on refining your balance and edging skills.
Rusty is on target with the "turning with the body" observation. Watch in your video how after your pole plant you drop the hand you planted with back behind you, then swing forward your other hand (the one you dropped behind you after the prior pole plant), as you go around the turn, preparing to make the next pole plant. With that swinging hand swings your shoulder, creating the body powered turn Rusty is commenting on.
That body powered turning method is called rotation, and It's a labor intensive method of making the skis turn, that results in something called a pivot, where at the start of the turn the tails fly out to the side as the skis twist violently in the direction of the new turn. The big start of the turn pivot is then followed by a big sideways slide of the skis through the bottom half of the turn.
Your first step is eliminating that rotation and pivot, and learning a more refined method of turning your skis. That method is called steering. In high level steering, the skis are turned with a gentle manual turning of the feet, there is no pivot, there is very little skidding of the skis sideways, the track you leave in your wake is very narrow, and the shape of the turn from start to finish is very consistent (in other words, no rushing at the start).
Begin by assuming a good athletic stance. Hips over your feet, not behind them, and shoulders over your hips, not in front of them. Once in that stance, focus on the bottom of your feet, and try to feel equal pressure on the balls and heels all the way through your turns. It's called being center balanced.
Then get on a very gentle slope (actually, do all of this there) and work on making long, gradual turns, being very patient to let the first half of your turn happen very slowly. Think of it this way; if your total turn is going to change your direction of travel 140 degrees, let the first 70 degrees happen slower than the second 70.
No shoulder twisting or arm swinging as you make these turns, and try to keep your hips and torso at all times facing generally in the direction your skis are pointing. If you you do that, and your skis pivot the start of your turns, you'll know it and feel it, because your hips and torso will twist downhill quickly too.
A good drill to be sure you have the rotation issue under control is called the HANDS ON HIPS DRILL. Leave your poles at the bottom of the lift, and ski with your hands always on your hips. Keeping your hands there controls the arm swing, which helps control the shoulder twist, which helps eliminate the pivot. It also lets you feel the rotational state of your hips with your hands, to be sure they're following the skis.
A way to make sure you are, and to begin to develop a movement pattern that will lead you into higher forms of skiing later, is to use your hands to gently drive your new inside hip forward as you begin each new turn. By that I mean, if turning left, push your left hip slightly forward at the same time you begin the turn. It will produce something we call counter, where your hips face slightly away from the direction you're turning, and will auto roll your skis onto their edges, so the skis can help do the bulk of the turning for you.
There are a lot more skills to learn to master this marvelous sport, but hopefully this will give you a means to jump start the process. www.YourSkiCoach.com