When I do the ground training, I start by pushing the shoulder, rear, or side to teach the horse that it should move when I push. When I start refining that to teach the horse to move on cue, I use either my thumb or the end of a training staff, depending on the force required, to get the horse to move by pushing on a point right where my heel would be if I were riding. In other words, if I want the fore end to move, I push my thumb into the horse's side just forward of the cinch. If I want him to move his rear quarters I push aft of the cinch, where my heel would be pushing if I were riding. For the sidepass, the pressure goes about in the middle between the two other cues, or about on the cinch. I teach this last. I do just what you did in teaching the horse to move by crossing the near hoof over the far one as they move. You should teach them to move without pressure on the leadrope, responding only to the cue. This teaches the horse how to do the move on the ground, so all you have to teach when you get in the saddle is the cue. The ground cue is meant to simulate the riding cue, to shorten the learning curve.
Once you are in the saddle, you have to start over again by giving the heel as the cue and looking for the first effort to do the right thing, then releasing the cue and giving praise. By doing this and progressively asking for more, the horse will quickly learn what you are asking for and will execute the move the way you taught him from the ground.
Now, when you start giving the cue from the saddle, at first you will have to check his forward movement with the reins, until he starts to "get it." Remember, a horse will only become as precise with his moves as you require before removing the pressure and giving reward. And he will only learn to respond to light pressure if the next time the cue is given is more uncomfortable. I give the cue once, twice, three times, then it gets progressively more forceful. The graduation in force is quite steep. They quickly learn that taking the first cue is better.
Consistency and firmness with the cues is paramount. Also, I find the proper use of spurs greatly increases a horse's willingness to be obedient and shortens the learning curve greatly. I have found the difference in the responsiveness of a horse during training when ridden with spurs and without is astonishing. As with any training tool, however, you must learn to use them properly and you must get your horse accustomed to them before just jumping on and jabbing.