In this video, I'm on the gray horse. I know he's sort of half trotting half cantering at parts, so it's something I need to work on. He's a pretty lazy lesson horse. Please give me any tips or critique you may have.
This comment has nothing to do with you, since you are on a lesson horse and do not own him. In my opinion the horse is probably not appropriate to use for lessons. He appears to have a physical unsoundness such as arthritis that makes him unable to hold a canter properly. I can see why they want to use him for lessons, because he seems gentle. But it is not fair to the horse or rider.
For you, it is important to keep your elbows bent but relaxed. I always try to remember that my upper arms "belong" to my torso. That means if the horse needs a longer or shorter rein, I should adjust it with my hands rather than pushing my upper arm forward or pulling it back.
Here is an illustration:
The reason for this is not for looks. If you have your arms behind your torso pulling, the reins will be very long and you will not have strength. If your arms are in front of your body, it will change your balance and also you will lose strength. Keeping your arms near your upper body means you can use your core strength when you need to on the reins.
Another thing I notice is that you are sitting on the horse as if he will never try to jump to the side or spook. Heavily on your bottom. You want to sit on your horse so you could suddenly stand up if you needed to, using the stirrups.
You should be able to stand up off the saddle without holding onto anything, and then after you sit, have the feeling you could stand again at any moment if you needed to. Having that weight down your legs against the horse's side is important for a secure seat during the faster gaits or a spook.
Notice that this rider's leg is not away from the horse's side, or pushed forward or back. The lower leg against the horse adds stability.
I think you are probably doing about just what is expected of a relative beginner. I agree that keeping your upper arm more in contact with your rib cage will help. Also, keeping more weight down into the leg is valid , too. I suspect your instructor talks about this. These things take time and repitition to develop muscle memory. I think that will come in time.
The horse is barely cantering. More of a "tranter" (a canter that hasn't really made it out of trot) but until you're established and comfortable with a more forward, bigger canter, that's okay for learning. I'd like to see more bend in your elbows, and also watch that your upper body doesn't lean in on your turns. Think of being a pole on a carousel horse. You don't need to lean in to turn, especially at these speeds. The horse will balance through the turns for you.
Thank you all. I don't love riding this particular lesson horse, and all of the other lesson horses definitely have a more forward canter, so hopefully I'll be able to get more videos of myself riding on one of them. The arm thing is something I'm trying to work on, and it has definitely improved from what it used to be.