You rode much better on Billy. He is a better horse for your level at this time. Your mare is a tough one to stay balanced over 'cause she kind of "pronks" like a gazelle with her head in the air and trying to run right out from under the saddle. I think those videos were taken before you changed to a better fitting saddle, no?
If you could take some lunge line lessons, on Billy, this would be absolutely invaluable. YOu are stiff from top to bottom. All that motion has to go through your body, and if you are stiff, you end up moving around a lot , forced to by the motion because you DONT move, you end up being moved.
If you are flexible in the hips knees and ankles, and allow your elbow to open and close while posting , you will absorb the motion of the horse, and though you will BE moving, you will LOOK still.