Some great observations already. I have no arguments (even with the conflicting observations!). Here are some observations of the same things with different words to see if they can help.
I love the angulation you are getting in your turns. This is upper lower body separation along the vertical axis.
Tip lead should match the angles of the hips and shoulders. At the finish off your turns, your ski tips are at a bigger angle (spread more ahead/behind) than your hips and shoulders. With your new inside ski trailing behind to start the next turn relative to your hip, your hip will be blocking movement to the inside of the new turn. This forces the vertical move to get up and over the blocking hip.
See how far your left ski tip is ahead of the right relative to the left hip versus the right hip? If we had less tip lead, it would be possible to make the next turn work better. But if we had the same amount of rotational separation in the hips and shoulders as the feet have (i.e. hips and shoulders parallel to the line between the ski tips), this could enable some powerful movements.
Look at where you are balanced against the outside ski. It's more toward the end of the turn than the middle. When you go up and then in, you tend to weight the inside ski more through the middle of the turn. Like this...
See how much weight is over the inside ski? Could you lift it off the snow here? What would happen if you lifted your left shoulder higher to get the shoulder line to be parallel with the fall line?
This is the same turn at the end as you are preparing to change edges. Can you see how your weight has shifted from inside to outside ski from the last still to this one? It should be the other way around. We want to start loading the new outside vs the old one.
The good news is that for the one turn you don't have excess tip lead at the finish of the old turn you look just like Tek (who is already level 3).
But the bad news is that this is essentially the same problem as above. The upper body is facing the same direction as the skis. If you were bending the new inside leg instead of the hips, you'd be able to get on the new edges quicker. How do you do that? It's called separation above. I'll call it steering into counter.
How do we do this? You referenced getting stronger in the core. I suspect flexibility and imprinting movement may be more of an issue, but it does not matter. Here are three exercises and a drill that can help.
Femur rotation - Lie on your back with one leg straight on the floor and the other raised 90 degrees (heel facing the ceiling). Turn the entire leg to rotate the foot through the maximum range of motion. Feel the stretch in the glutes. Try it with the leg as straight as possible and with the knee slightly bent. Repeat x times with both legs.
Medicine ball - Stand holding a 10-25 pound medicine ball with both hands in front of your belly button. Step one foot outside of shoulder width, then shift your weight over that foot as you rotate your torso in that direction. Keep the ball over the belly button so that do not over rotate the shoulders relative to the hips. Repeat x times in both directions.
Rotational weight shift - Stand with one foot advanced so that the heel is even with the toes of the other foot. Start with your hips square to the direction the toes are facing. Rotate your hips so that they become parallel to a line drawn from big toe to big toe. Feel the weight shift onto the lead foot? Imagine that happening as you complete the old turn instead of having tip lead decrease as you finish the old turn. Start the new edge change from the point of maximum tip lead.
Picture frame funnel - hold your poles in the middle of the pole instead of the grip so that the poles provide a "frame" against an object visible at the bottom of the slope. Start making narrow radius turns without letting the object get outside of the picture frame. Make increasingly larger turns until you are no longer able to keep the picture in the frame (ideally until you are finishing turns slightly uphill). Be careful about crowds while doing this.