"Toy hauler" isn't a type of hitch, so your first "vs." is incorrect and misleading. I don't know if it's a real term (or if it's slang), but from the pictures on the site, it refers to a trailer that is used to transport your "toys" -- one of the images on the site shows a race car or similar inside the trailer. The "toys" could be all sorts of things, including a snowmobile or other equipment. Perhaps the category of "toy hauler" could include trailers with spaces for horses, but horse trailers are subject to special design requirements, and not just any "toy hauler" would qualify as a horse trailer.
I've seen these things at camping shows in the US. They can be huge! As the images on the site show, there can often be a full-size living space in the front part of the trailer.
A fifth wheel is a "Sattelkupplung" or "Sattelplatte" -- the kind of hitch used by semis. It's technically the plate that's on the back of the cab. "Fifth-wheel" trailers are called that because they use that same type of hitch, and the fifth wheel is often mounted on the bed of a pickup truck. The site you linked to shows such trailers.
Taken literally, I'd say that "bumper pull" would refer to a hitch that was (temporarily) mounted to the rear bumper of a vehicle. I think that various safety-related design features have made that temporary mount obsolete, so that the hitch is always permanently attached to the chassis of the vehicle now. It's that standard type of hitch that is used for pulling various types of trailers and which is very common in Germany.