Proprioception has been a main theme for me for a number of years. I'd say it's not just movement, but rather, a sense of your body in space. Movement would be part of that. If it were only a matter of being able to play the piano without looking, with your eyes closed, then I'd be all set. Originally that's what I did because I discovered I've always had a kind of LD where some visual things confuse me - directional, and symmetry. On the piano I went by the higher sound being "over there" (sound and touch) rather than "visually 'there' " related to visual keys, and "to the left / right". But it was paired with awkwardness because of the poor sense of body in space.
When I was in an "adult violin students" FB group, we heard from someone who taught one aspect of this. I'm sorry that I didn't continue following. She had us consider the space behind us and not just what's in front in our visual field, and for some reason that made a difference or was intriguing.
Proprioception for some may also involve your body memory vis-a-vis things when you were a different height and proportion (child) and catching up to where you are now. Or, when you're playing piano, do you have a sense of your feet and body as you play and move from this part of the piano and that? Most instruments are smaller than us - the piano is bigger so we are the ones moving around on its surface.
I had a posture of major lordosis, which also meant that how the arms and such moved was all a compensation - new ways of moving had to be remapped in a few cases. That is probably not proprioception. But a certain way of jamming down the shoulder blades meant my elbows (and sh. bl.) were lower than normal and I had to change that body map. That also affects movement in space, and where you think your arms and hands are vis-a-vis your surroundings, and thus how you move. Most people won't have something like that.
navindra Do many piano teachers even recognize the importance of proprioception or explicitly teach this as a skill?
I've seen it in two places so far.
Seymour Fink, "Mastering Piano Technique" - I have the book and the CD. In one section he has you play particular intervals in areas of the piano so as to get a physical map of where things are.
Alexander Woroniki - I think he addresses something similar.
I'd also say that the teacher or book that keeps you in the middle of the piano for a long time, mostly C, G & F "position", five-finger spans, no pedal ..... they will be working against developing that sense of body and space and "terrain" on the piano. It sort of cripples what could develop.