A horse should always have protection on their legs because things happen and better safe than sorry.
Polos, brushing boots, open front boots, hock boots, jumping boots are kind of standard choices in protection.
Over reach or bell boots are more about preventing problems like the horse kicking themselves from behind or stepping on a shoe, etc.
I recommend at least one or the other; sometimes both are combined for extra protection.
I have brushing boots for my horse and I set of polos which I don't really use much anymore but gone are the days where I ride or lunge without protecting my horse's legs. He is my investment after all.
For anything to do with jumping, you need to protect your horse's legs. Open fronts I believe are what most jumpers use. Jumping boots are used for XC/eventing/jumping as well. It all depends.. gotta find what works best for your horse and what event you're doing.
"Strength is the ability to use a muscle without tension"