For the high jump, I think the winning technique in the near future will evolve into a side hop with a tiny unicycle. It will have lots of tire, pumped up high enough to give you a good bounce, and a small wheel to keep the weight down. If such a small wheel is used, it will have to be a side hop.
For a not-so-small wheel, I can picture forward-hopping with the seat in front. It should be possible to build up some speed with seat in front, and hopefully this will make it easier to clear the bar when you reach the necessary height.
I think the reason Dan Heaton prefers the forward hop is because he has polished it and been doing it for a long time. But if he focuses on side hopping, I suspect he can get higher that way.
The competition event name is High Jump. This is because you can say high jump and everyone who ever watched a track and field competition knows exactly what you’re talking about. If they’re not sure, you can tell them it works the same way as an Olympic high jump. That’s why we call it the high jump.
For a jumping competition where you don’t have to clear a bar, a different name can be figured out, but it won’t correspond to any track & field event I’m aware of.
We call the Long Jump the Long Jump for the same reason. Side-hopping in the long jump is not going to be the way to beat the forward speed hoppers. There’s a limit to how far you can do a side hop, and even a slightly smaller limit to what your wheel and tire will hold up to. I think the best results in this event will be high speed forward hops.
The best unicycle for this event needs to combine three relatively incompatible features. You want speed, which means a large wheel and/or short cranks. You want a good jump, which means some bounce in the tire, which suggests a high air volume tire. And you want the cycle to be as light as possible, to keep it in the air. That’s a tough combination.
If it can be found, possibly a 24" wheel with a fat tire will work well in this event. Then you’ll need a great technique and timing, to match your higher speed with a perfectly timed jump and series of movements to land you, riding, on the other side.