Yep! We were learning the one-rein stop. My trainer taught me that one hand should be kept on the whithers to stabilize yourself and also help your legs relax. The other hand should reach down as far as you can and pull your pinkie to your belly button. The horse should eventually give to the pressure and relax his mouth/head/neck.
Clearly, my one hand pulling back was a flat against my stomach and this picture was taken before Weiser relaxed, which could have something to do with it, in hindsight. I also don't think I reached down far enough that my elbow had to jam back too far in order to make his neck turn to the side.
For the sake of the exercise, we returned him to neutral, if you will, once he completed the task of stopping and relaxing. Honestly though, Weiser is such a steady sweetheart that he is a gem to learn on and make mistakes with.
What kind of bit was being used? IMO, a double jointed, thick-ish (comfort) , Snaffle is the only bit I'd use to teach a horse or a student the one rein stop. It looks like a 0-ring snaffle, but no way for me to know if it's double Vs single jointed. A single jointed one could be poking him in the roof of his mouth, or having a 'nut cracker' effect in his mouth. Maybe you can discuss that part of it with your trainer?
I'd also suggest that you clarify with your trainer if Weiser already knows the one-rein stop, and is just testing you or uncertain of your cue ,,,or if you are 'teaching' it to him. While I don't see a problem with the student learning a one rein stop, at the halt to begin with...I hope your trainer doesn't have a student trying to 'teach' it to the horse (as others have said here). I further think it would be helpful to you to learn it from the ground first and then progress to learning it as a rider. Your instructor should know how to have you do that as well.
Whether from the ground or riding, this is just a easily/effectively done in a rope (ie Natural Horsemanship) halter, which would alleviate your concerns of being too harsh in his mouth.
Yes, your technique would be easier, and probably more clear to him, if you reached further down the rein so that you don't have as far to pull. Go slowly, this is a simulation of an emergency, right?, so no real hurry....actually close your hand slowly (one finger at a time will give him warning of a cue coming) over the rein, then turn your fist so that your fingers are 'up' (another cue to tell him something is coming), then---I differ here from what your trainer is teaching..so take it with a grain of salt...I don't want to cause confusion...I slowly bring my hand to my mid thigh and place it there and hold until he softens his 'give'/face to it AND stops his feet. He may go in a circle-ish until he stops moving, so be prepared to feel like he is spinning. He's not, but it feels that way to a beginner. For what it's worth,,,I don't go to the bellybutton (or to my opposite shoulder) unless I am asking for a hind quarter yield with a horse who is learning it (HQ yield from the saddle).
Good thinking to be wondering if you could be hurting him in the mouth !
Good luck. Have fun...stay safe!