It sounds like you're thinking about a number of related things ....
a) You sensed yourself feeling and acting defensive as you skied.
Have you skied with less defensiveness? If yes, tell more; let's figure out what can lower that feeling.
b) You are aware that the distance between your skis' tails is wide, and that the distance between your feet is wide.
Do you feel this as you ski, or did you only notice it in the video?
c) You connect the wide ski tails/wide foot stance with the sensation of defensiveness.
The last time you skied with narrower feet/skis, what was going on that helped you do that?
c) You have difficulty getting your new outside ski (uphill ski) to start the new turn because it is so far outside of your hips.
Correct diagnosis. I can see this in the video. You are pushing that new outside ski out farther at initiation, and then because that foot is then so far out there, your body braces against it. This works to start your new turns ... but but but ... it's a harmful habit and needs to be replaced now. So I agree, narrowing the the wide stance and wide tails is a good goal.
d) You did not feel tongue-shin contact in your boots, but you have in the past.
Do you remember what you did to maintain tongue-shin contact in the past?
e) Your hand position in the video worries you; you think your hands should be farther forward.
Your hands do not need to be farther forward; focus on feet and legs at this point)
f) Your turns lacked the rhythm you've experienced in the past.
I can see that in the video. Easy fix: try singing in your head, and turn to the rhythm.
g) Your instructor asked you to lift the new inside ski; you had some success with this drill at faster speeds, but not as much success as you'd like.
Were you attempting to lift the whole new inside ski, or just its tail? Just its tail is enough, leaving the tip gliding along on the snow. If the tip rises, close your ankle to get tongue-shin contact and try again. This is a good drill because it tells you if your'e aft or not. .............
But first you might try skiing in a narrower wedge. Close your ankles for shin-tongue contact, slide your feet closer to each other, and make gentle wedge turns with singing in your head for rhythm.
h) Your instructor had you doing lots of drills to try to get the skis parallel
Which other drills? Any success with them?
i) You had some good runs; you felt mostly in balance, and despite a near fall, you stayed vertical!
Can you focus on the good sensations these good runs gave you? Seek those good sensations in every turn you make; that's why we ski.
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My overall thoughts: I think you are way aft, which causes insecurity. A really wide stance is a way people try to get a more secure feeling. It doesn't work. The wide stance messes up turn initiation as you've experienced, and it forces you to stay aft. So you're right, you need to narrow your stance.
To do that, close your ankles to maintain continuous shin-tongue contact, slide your feet closer together and keep them there, then ski in a wedge. Don't lift a ski at first (that's scary and destroys feelings of confidence!), sing in your head to get some rhythm going, loosen up and relax and enjoy your runs. Find your bliss. Your confidence will return with that bliss. This will all come back to you when you can relax (you are very rigid) and ski rhythmically without worrying about doing new things. Eyes on the prize: bliss!
Once you can make wedge turns while relaxed, you'll be centered (not aft), and occupying that wonderful state of skier euphoria. Then you'll be ready for getting parallel with all those fine drills.