It isn’t so much a case of teaching the horse his leads as it is teaching him what it is you are asking. One thing that can help is to have someone put your horse on a longe line while you are on him (assuming he takes the correct lead while longeing), so that he can associate your cues with the action you want. Horses are born with and use the ability to take either lead and do flying changes while loose, so what you want to teach is the response to a specific cue. And, you really can teach the words “left” or “right”, as well as “canter”. You can even teach the word for a specific place in your ring. Of course, you cannot be loud about it, but horses are capable of learning a huge vocabulary. They can even understand (but not always obey) the command “get your head out of the garbage”.
For specific cues, G&S is correct.
It is never the horse's fault