Jumping video, I have a hard time seeing your position, but from what I can see you are staying out of his way and not lying on his neck, both good things. The one criticism is that coming into the fences, it looks like you are almost bouncing, which makes me think you need to work on strengthening your lower legs and balance.
On the trotting video, I'm going to break it down into body parts
upper body: you need to close your hip angle some, and let your horse post for you. You are doing what I call pelvic posting, meaning you are leaning with your pelvis and pushing yourself up. Your horse should be pushing you up, you should just be controlling that movement. Also, shorten your reins and try to keep a steady contact with the bit. You really aren't helping him much with your hands.
Also, work on your diagnols (sp?)
lower body: your legs look pretty still and quiet, which is good. However, they could be more effective, which would help drive your horse up into your hands and help with your post more. To make them more effective, think about putting your calf on the horse and letting your knees come off. Right now you are pinching some with your knee, not bad, but it's keeping you from really using your legs well.
Hope that helps some, and I'm glad to see a trainer using a gymnastics type of exercise. It will do wonders to help a horse and rider.