That's my point. I consider a burpee to include the push-up and the jump. But not everyone does, as you proved by saying it doesn't include the push-up. And just yesterday I followed a link from one thread on here (Don't ask which one, I don't remember, but I think it was an article on T-Nation) that described the burpee as being the same as a squat thrust, but with a hindu push-up and a regular push-up, but no jump.
I'm not saying who's right or wrong, I'm just saying that it depends on who you ask. The standard around here though, as Biggiebrother pointed out, seems to be with push-up and jump.