Much agreed with Muppet.
I suggest yourself, your partner and your daughter take an opportunity to watch a large group of horses interacting in a paddock environment.
You will find that it takes ALOT to physically hurt a horse. My 2 year old has been booted so hard in the ribs by the dominant mare in the paddock, that it sent him flying onto his side. He got up, shook himself off, and never tried to push that mare around again!
They kick each other, bite each other, slam into each other, rear and strike at each other. A human slap feels very insignificant to a horse. Unless you start beating them with a steel rod, it's **** hard to hurt them with your own strength.
The issue with physically reprimanding a horse however, is that a horse is a heck of a lot stronger than a person, particularly a 13 year old girl. We need to use brain, more so than brawn. You want to shock the horse into moving away from you, and thinking that it better not try that again. I prefer to use my elbow if the horse is leaning into me, I leave my elbow out, and if the horse wants to bump into me, it cops an elbow to the ribs. As soon as it moves away, it doesn't have the discomfort of the elbow, and learns to stay away from your 'bubble'.
When correcting these behaviours, you need to be quick and sharp, then go on your way as though nothing happened. Horses learn through the release of pressure, not pressure itself. So if you apply pressure, and don't release it the SECOND the horse responds, the horse will not associate its response with being correct, and will probably go straight back to leaning (or whatever other unwanted behaviour) again.
Ground work lessons will be very beneficial to your daughter, and may I suggest that yourself and your partner attend, and be willing to keep quiet and simply learn?
I remember when I got my first pony, my mum thought a pony was simply a big dog - and reacted in similar ways to your partner towards discpline. If we treat a very big, and potentially very dangerous animal as though they are a cute fluffy pet, then we are setting ourselves up for injury.