Yes. That's a good explanation. Especially about having your body mirror your intention. If you want him to go, there should be "go" energy in your body. YOu stand more upright, your chest sticks out a bit, you kind of have a "lift" in your step. For the "slow down" you do the opposite, as OHL said, you kind of sink into yourself and emphasize the outward breath. It's you showing through your body the intention in your mind.
If you want him to canter, when you ask for that, have a real committment in your mind of "Canter!" and raise the energy in your body . HE willl feel this. He may play up a bit, essentially tantrumming, but that does not matter. Just stay true on your course of directing him to do what you asked. His tantrum is his releaseing of the resistance. Let him release it if he must, but do NOT stop what you are asking just becuase he is upset. That will only teach him to play up and tantrum MORE. Stay calm and go through with the request. You just keep asking , no harder, no softer, until he move on. Then like OHL said, continue as if nothing happened.
I think you should learn how to "send" this horse places. It's basically making him move away from you., and to where you direct him. This might be considered a "game" by some,. Since Parelli uses it. I don't 'know what he calls it, but you start sending the horse over to a barrel or a stump or something, and you can put a small treat there. (put it there ahead of time) you don't treat him by hand, and you dont' treat very often, but the idea is to sharpen your ability and his willingness to be sent away from you but still be mentally "on line".