I don't know the name of the type but a model is 'S-Kite'. I didn't realise you could get huge versions of trick kites, and they give lift..

How are foils different to them when jumping or getting air?

Examples:





Pete

IMHO, I find that the two line delta stunt kites are easier to control, but if you're not paying attention to them they're easier to crash too as they move fast. Mine isn't nearly as big as the one you have in the picture (it's a mere 10' wing span but gives off some pretty good pull when the wind is kicking). I've had the kite for almost 8yrs. now and have broken more spars and one small rip caused by a broken spar puncturing it that I had patched. Foils just seem more durable overall and easier to control while not looking at them. I can easily see how a foil could out-live a delta stunt..ecspecially if you're jumping with them.
Just my .02 ~Joe