I always start out teaching a child that age to ride using two hands (direct reining just as in driving) and a snaffle bit so that the rider learns how/where/when to move the horse and stop the horse. That way you are also only working with one at a time on training something new to.
I also teach the child how to hold the reins with both hands, with one hand and using the other to reach down to pick up slack for directing to the side (and they learn to do it with both hands) and just holding the reins with one hand while holding the mane or the saddle horn with the other (and again swapping hands on the reins). Also, the pulley rein, one rein stop or emergency stop (all the same, though called by different names) is taught as is how to dismount a runaway.
Here is older (9 or 10?) Jasmin's 2nd riding lesson. She has graduated off the lounge line and is in the open since I don't have an actual arena.
3 lessons later - back on the lounge line and doing more rein exercises.
and riding off our property on a 2 mile walk about...
In September, Jasmin did start doing some neck reining. Most of that work was done in the round pen only as she started grasping the concept that she wasn't to pull while neck reining but only to lay the rein on the pony's neck between the saddle horn and her head.
Magic does have the general concept of neck reining herself, but she was mostly ridden as you see in the pics above with a snaffle and 2 hands by children only beginning to ride, so not much. She never was introduced to a curb or shank bit and was just as often ridden english as western.
Laura was a couple of lessons behind Jasmin due to being sick and not coming to any lessons at first. Then she had major fear issues. We spent much of her first lessons with her just mounting, sitting, dismounting and getting comfortable with the pony. She took much longer to graduate to any rein use, simply because in her fear her body would go rigid and she'd snatch the reins up to her chest and lock up.
But here is a good pic of her grasping the turn around and yield the hind quarters to the left while turning right...
2 months of 3x week riding lessons and Laura still often locked up. While the stirrup irons could have been dropped one notch, she does have enuff length to get her legs down and back, but she's drawn her legs up as well as her hands/arms. She was practicing her turns - the bucket was a focus and she was to ride forward and around the bucket. It was a hugely frustrating exercise for her as she tried to get everything working together while handling her fear. She did worse when in a western saddle. Now, 5 yrs later, Laura no longer rides at all, but will go for rides in a cart or a wagon. She has no desire to learn to drive herself, though. She does sometimes groom and do ground work with the ponies her grandmother owns. Laura would have been 7 or 8 here... Not sure.
Stuffy does well neck reining with a snaffle and in either a western saddle or english. I never purchased a shank bit - was getting ready to when the above girls quit riding. I may get one later for her - I just leased her out again for a young rider. Right now, she's learning how to ride using two hands.
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I am a little bit different than PoniesRule, in that I use the right at the girth on both sides as "forward" and behind the girth to give sideways or yield, with the reins giving direction as to whether it's the front or the rear end that was disengaged. BUT since doing more NH type riding, I drop my leg further back to disengage the hindquarters, use slightly behind the girth with both legs as a "go forward" and right at the girth to disengage the forequarters. I never got into the "reining thing" of using my toe or boot heel in front of the elbow or on the shoulder.
That said, I found the 9 yr olds I worked with not able to reach very far back w/o unbalancing them forwards onto the front of the saddle or the neck. So I teach them to use both legs at the girth for forward and one leg just a little back of the girth (and it takes a while for them to grasp that concept) to yield - and use the reins to direct them as to whether forehand or rear quarters.
AS to teaching the neck reining to your mini - I agree with all above as to how to go about it. Since yours is so well trained already, it shouldn't be difficult at all to teach her to move over by just laying a line or rein against her neck. Most of my advanced ponies will "neck rein" from either side now, since I often lay the lead against their neck as I ask them to yield away from me. I use a tap with either my hand or a solid object (a short crop or a NH "stick") to increase the pressure until they respond to just the lead or rein laid on the neck. That "tap, Tap, TAP, whack" is a very effective training tool, LOL. Once they learn that the light cue is followed by progressively stronger ones, they get it quick!
Once your mini yields with the lead rope/halter, you can introduce neck reining with a bit and I think you will be quite surprised how easy it is. The hard part is teaching the child that it is literally just laying the rein against the neck - not pulling - which actually pulls the head the opposite direction of what is desired. I still sometimes have issues with that one myself and I've been riding western and neck reining since I was a kid. I like using two hands and a snaffle - but then again, I've spent most of my life starting green horses, not taking them all the way to finished riding horse. Most of our shetlands have more training than the riding horses I've started under saddle before returning them home or selling them.
Hope you don't mind that I prefaced this post with pictures of direct reining and learning to handle the reins. To me, it does pertain to learning to neck rein (for the child)....