You can also add one accessory to your boots its called spurs. Spurs are metal fixtures that are supposed to be connected to the back side of the foot, outside of the boot. This makes it easier for the rider to kick the horse in the belly since they would need less effort to make the horse feel the kick.
Either kind whether low cut or short cut is good. They both do the same things. Both leather and have a little bit of a heel. The only thing is in formal equestrian you need long cut boots since its part of the attire.
Here are some samples of high cut and low cut boots :)
Low cut slip-ons (garterized)

Low cut shoe lace

High cut with shoe lace & zipper

High cut only zipper
