Concerning the rising trot I don't know if I quite understand the purpose. I guess there is pressure in whatever place the stirrups are hung if the rider puts more weight in them when rising. I only see this as an advantage because that way the pressure shifts from one area (where the rider sits (or stands as Josepha puts it) to another, stirrups and rider's thighs. Am I missing something here?
Most people post too high.
One should barely leave the saddle. This better insures the rider is in time with the horse, and that the horse has the maximum ground contact as the riders' weight come back into the saddle. Stirrup pressure increases the least when the rise is very low and of shorter duration.
This low rising to the trot allows the horses' body to be more in control of the action with their own mass motion contributing more to the rising and falling of the riders' mass.
Obviously too, the amount of "contact," time is greater, meaning the horse retains more "feel," of the rider and can receive more information including those bits of information that constitute cues.
In fact, when you have gained some skill with low posting you can prove to yourself that the horse is better treated by this method by dropping stirrups and posting without them. A lot of childrens' equitation classes include this little exercise in horse shows. "drop stirrups, post to the trot please."
It's standard in my own instruction. If I ever get back in riding shape again (long layoff recently) I expect it will be my standard measure to determine if I am in shape yet.
One does not rise only or predominately from the stirrups, when posting is done properly. One moves their mass with the help of the horses' movement of its mass.
The picture I want my students to have in their mind when posting is that of trying to hide the rise .. to make it so small it appears to be something the horse is doing, not them. I doubt one can, in real life, do that, but it's a good goal to aim for.
Finally, I question the assumption which presumes that posting is less impactful on the horse than sitting to the trot.
A rider that develops the ability to sit a trot, including an extended trot, as dressage riders do, is, I believe, barely impacting the horse at all, whereas unless one is the most skillful of low rising posters posting itself is impactful and may even be detrimental.
An opinion, mind you, no scientific backing for it other than examining the horses' dynamic anatomy, (being in motion) and through the laws, and rules, of physics.
Donald, Altea, and Bonnie Cupcake