Haha, that's a difficult question to explain. Even though they're different styles its a "genre" that goes together that's separate then say, eventing or the AQHA stuff. The main governing organization that runs it all is USEF and most shows (with the exception of some schooling shows) will follow their rules. The H/J shows will almost always have hunters, equitation, and jumper classes... and nothing else. I don't know many people who show regularly in the HJ world who show in other "genres". Once you go to HJ shows there is a big difference between the hunters and the jumpers and you'll think of it as 2 separate worlds there. :)
Yes, you are right about the idea of the hunters ( based on the horse: nice pace, good form over fences, nice mover, etc) and jumpers (fastest time with the least number of faults) Equitation classes are a part of it too. They're kind a mix between the two. You're judged on your form and how well you handle your horse. So you're supposed to go nice and easy like a hunter, but your course will be more technical to navigate and often have difficult turns and such. Most people who do hunters will also do equitation.
Not everyone who does hunters does the jumpers.
Hunter fences will look like something you see out in the field. So you get shrubs, flowers, brick walls, etc. The jumper jumps are made to scare your horse a little! Bright flashy colors, striped poles, scary standards, etc.
The Working Hunter division at USEF governed shows is a 4 foot division that has some tricker parts to the courses. But a lot of divisions are called "(blank)" working hunter. I'm not sure why. :) Like Green Working hunter, There are other divisions called "blank" Conformational Hunters. I think the Conformational is not just based on movement but also conformation (hence the name) and you have to model your horse after the class. Don't quote me on that though!