I know you said not to critique your riding, but it is difficult to just critique the horse and NOT the rider when the rider totally influences how the horse moves and acts. In this case the horse would move much better if you, as the rider, sat up straighter and helped drive him forward.
I like him better in the top two photos than later. Part of that is that this horse needs to carry a certain amount of weight to look good. He has gone from being a tad over conditioned to being under conditioned. See the lines along the buttocks from the top of rump running along to above the musles and tendons along the back of the gaskin? That line separates the semimembranous and semitendonosus musles and old tymers called those lines "poor lines" meaning the horse needs more groceries.
Some horses carry this line and are not "poor" but others, like this boy, do not look well when poor lines are showing.
Notice as well that in the top photos while the horse is not balanced and "poking his nose out" he does not look too bad. His neck looks normal but untrained. In the lower photos he has actually developed a Ewe Neck and is thin enough to show how long his back is and what may or may not be a "hunter's bump" on the top of croup.
What I am seeing here is a fairly untrained horse that is a pretty good boy in the top two photos that is developing over time into a horse that hollows his back and uses the muscles on the underside of his neck in the later photos.. IOW's he is developing exactly opposite of what you want to see!
Part of the reason for this is his rider who is using hands to make the horse into a frame and not using her body and legs to drive the horse forward and then slightly resist with the hands to get the horse under himself behind.
It also appears, from the musculature of the horse, that he is not doing any trotting up hills or over caveletti with a long rein, allowing him to stretch his top line and tuck his abdominal muscles. This work requires the rider to stand slightly in her stirrups and lean slightly forward giving the horse rein so he can stretch his top line. The horse is not being used so that he can access his 'elastic ring' of muscles.
The conformation of the horse is not helping either. He has a high butt and a long back.. both work against getting him round and working off his hind quarters. That being said there are many horses with long backs and high butts that do.. so training can over come a lot.
Beyond that the rider needs to get her elbows in close to her sides and to have a deeper heel and seat, sit up straight and try to roll her thigh into the horse better. This will not be easy for this rider, but is doable with practice.
Again, I know you did not want the riding critiqued, but the riding is what makes the horse move or train like he is, so it is impossible to speak to the horse without speaking to the riding.
Here is an article I recommend you read. I think it will help you understand what you need to try for with your horse. I have put a lot here and it is not easy to understand from the written word on a forum (especially the words I write!). Riding the "Elastic Ring" of the rib cage