So I'm guessing you will practice this from the ground until backing and halting are fluid and soft. So then when you do get into the saddle you would then use the same 'see-saw' (I know what you mean) motion? Or does it change when a bit comes into play?
Where I think a lot of folks get off track with early training is that they are pretty good riders and so they have been coached to have a pretty good form when riding.
In other words the horse has less experience than the rider is with the cue.
Now the horse knows how to be a horse pretty good and he also knows how to move pretty good from running around in the pasture.
He just does not know what that cue means.
In the beginning the Handler and rider have to kind of exaggerate the cue and balance to help the horse learn what they mean and as the training goes on that cue is refined to where the horse is.
To answer your question,I try to ask in a way that the horse will understand and get the idea of what I am wanting him to do.
I always try to build on what I have asked in the past and so I this case I would kind of stack the deck so he can get the idea even better.
When a horse is having a REAL tough time getting the idea,I do alley or chute work.
A chute is made out of a couple of panels and is wide enough so as not to bang the riders knees.
The horse is led through in both directions and then mounted like this.
Now after the horse is very comfortable(days,weeks,whatever)one end is closed off.
Ride the horse in and just sit there until you feel the horse wanting to go back.
When that moment comes add the cue and encourage.
Some horses need to really go left,right ,left,right when backing up and what I mean by that is that they just don't get the idea of back
being one movement.
They kind of see it as two movements or more and the handler starts by just getting a left foot only.
Then the handler wants the right foot only and that is the back and forth that you are talking about.
To get a good stop,then ride the horse forward to the backup.