I'll focus on the english part because I am an english rider.
The 4 gaits in english are
At the trot you can either sit or post. Posting is when you rise up out of the saddle everytime their outside shoulder goes forward. When you change directions, you sit 2 beats and when you rise again, you are on the diagonal. Diagonals is what you call the posting. As in, when you are trotting around, you are either on the correct diagonal or the wrong diagonal.
Here's a video of a posting trot. You can see as the horse's outside leg goes forward, the rider comes out of the saddle
There are also variations in the trot. You can have a sitting trot (which is usually slower than the posting trot), a collected trot and an extended trot. If you watch dressage videos on youtube, you'll see a difference in the horse's speed. The horse will also cover more ground with the extended trot.
Here's an example of a sitting trot. I was taught to make yourself like jello as you want to follow the movement of the horse
Here's an example of an extended trot
When you canter the horse, the leads are the same as in western. If you are going to clockwise, the horse would be on its right lead. If you are going counter clockwise, the horse would be on its left lead
Here's an example of a canter. At first the horse is on its left lead and then at about 13 seconds in, the horse does a flying lead change, changing to its right lead
Just like the trot, there are different variations in the canter. Working canter, collected canter and the extended canter.
I hope the videos helped