Well that last video shows a pretty nice ride. The horse is reasonabley forward in the canter and you sit really beautifully and quietly. I envy that. I think you guys are a nice pair.
When you are trotting, the loop in the rein and your hands curled under makes for a much more inconsistent contact.
Working with helpin this horse follow and accept contact will help a lot, but for you to do that, you must offer him some solid contact. I know he curls away from it, so you kind of "throw" him a few inches with your wrists, but instead of that, or at least along with giving him a couple of inches of leeway, put your leg on firmly and get forward. when the horse curls behind the contact, the ONLY answer is more leg, more forward. You can open the front door at first, and once you have him more responseive to that foward push , you start not openning so much ., but having him move into a steady contact, so the only thing that changes is the amount of drive he is putting in.
Raising your hands a bit when you open the door and put your leg on will also encourage him to bring his head up and not dive behind the bit.
If you need to shorten your reins to maintain contact with him when a squeeze of the hand is not enough, then you can widen your hands apart from each other to make the rein effectively shorter.
Seriously, I think your ride really well and have such a nice solid leg. Once you get your hands straightened and get him to move out more, you will be a competitive pair.