I see some improvement from that last video you posted for critique. How long have you been taking lessons?
I think I agree that starting hunters is your best bet. Although jumpers are not judged on their seat, they need to have a seat that is pretty strong, even if it's not "pretty", because they GO!
You still need some work on the stability, and like another person mentioned, it seems to be the core. Your body's core. I think that what happens is that you tend to ride off the ball of your foot a bit, and maybe are a bit stiff in the hiips so that when the horse canters , instead of going with the motion and absorbing it, you get kind of tossed around .
I would have you do lungline lessons, sitting in the saddle , no posting, no stirrups and no reins. Or stirrups only, no reins. Just work on following the motions of the horse by sitting and letting your hips move with her.
You do keep a nice upright body and head and your hands have improved a LOT.
Bee is a sweet horse!