First thing - contact!
Can you see that you are getting the old 'loop-tight-loop' effect happening in your reins, especially at trot? This is coming from a) your hands moving and b) reins are too long.
Shorten your reins, and try riding either with a neck strap or hook your pinkies under your saddle cloth. By fixing your hands, you will realise how much you want to move your hands!
Because of the suspect contact, your lovely horse is a little hesitant to come into the bridle for you. He is trying, so if you hold up your end of the deal and give him a nice consistent, constant contact, he will reward you will coming nicely into that bridle.
Transitions, transitions, transitions. They are SLOPPY, and I know you have the knowledge and skill to do better!!!! Sharpen them up, it's very easy to get lazy with them I know ;)
Walk to trot, should be snappy, immediate trot off with energy. Not almost 10 'jog' steps to get into a slow trot. Ask once, and if you don't get a snappy reaction, do not be afraid to give him a flick on the bum. He needs to be so much sharper. Ride a big trot, little trot, medium trot, little trot, big trot and so on. Expect the transitions within gaits to be immediate and really sharp. Doing this will improve the overall work out of sight. Don't let him be lazy, but don't nag him. Expect him to stay at the same pace and temp within that pace until you tell him otherwise. If he wants to change, give him a quick reminder then settle again and just let him be. I was getting dizzy and a bit bored in the first video watching him trot round and round and round to the right with no change in tempo, no test of the inside leg to outside hand etc. Don't do more than 10 strides that are the same. Always readjust, ask a little more, a little less, a bit more bend, a bit more flexion, a bit of counter flexion, counter bend, can you move him off your inside leg, can you move him off the outside leg, will he move his shoulders in or out, can you move his haunches, can you put him a little deeper and rounder, can you then ask the trot to come bigger while staying deep and round...??
That is what you need to be thinking every.single.time you get on this horse!
As for your rider, I'd really like to see you drop your stirrups a hole or two (I know I know, you're an eventer ;) but for your flat work you need to drop your knee a little more). At the moment you are balancing a little on your knee and thigh like you would if you were out jumping. In flat work, you want to release your knee and allow the leg to hang. You barely need to come out of the saddle in rising/posting trot. Just scrape your 'front bum' along the saddle and sit again. We rise to help the horse open his back and keep balance if he is not strong enough through the back to hold you in the sitting trot for long periods. There is no need to come far out of the saddle except in specific exercises where there my be a little more daylight under your backside exaggerating your rise.
You really want to stay as close as possible to the deepest part of the saddle, as that is where our centre of balance is.
In the canter, you are coming significantly out of the saddle. Just sit girl, you're making it harder than it has to be. Sit deep, put a tissue under your bum next time you canter and keep it there! Allow your legs to just hang. The only movement should be through your pelvis. Your core needs to be doing all the work to keep you balanced, think of hula hooping, it is a similar action.
Again with the hands, keep them quiet. If you need to rest your knuckles on his wither or neck, do so, just to give you a feel of how the topline stretches and contracts.
You will probably find it easier to sit the canter if you choof him up a little, at the moment it's 'nice' but very pogo stick. Put your leg on, sit on your bum and ride his wither up and forward.
That said, the rhythm is good. Keep that, but bigger and more expressive!!
I don't have time to watch the 2nd and 3rd video at the moment, but work on the above and it will help iron out any issues in those movements also.
You two look lovely together and you are doing a really good job with him. I'm just being a nit picking so and so, because I KNOW that you have the knowledge and skill to do it :)