It might take you a long time to get from the first video to the second video. Especially if your horse is naturally trotting like that.
You can use very small circles and lunging with a tie down or side reins.
Also, Jazzy was right about using your seat to help you. Western people sometimes post when training, as it is incredibly useful to the english people. We just don't get to show with a post.
It might also help to collect his mouth very far underneath him and keep the leg pressure constantly on him so he doesn't break the jog. Before you do this, you should probably teach him to maintain his gaits better by half-halting like you said, but not allowing him to transition down. With both of these ideas, remember to focus on bringing him into a relaxed, somewhat extended jog. You're basically trying to show him that he can relax and level out without losing the two beat jog. When you release rein pressure he needs to immediatly relax his neck into a lower frame and continue forward with a slower rhythm.
Hope it helps, yours is a strange case!