it originally comes from Surfing. It's not an "off the lip to air" either. It's when you are riding a wave (fairly fast up the middle of the wave) and hit the lip as it curls into white wash as you get air (timing is everything). Then you come back down into the wave/white wash...Which is a re-entry. Not sure I explained it the technical way, but that's the way I knew of while surfing and growing up in Hawaii.
Yes, it's all the above: Speed, Ollie, and hitting it just right. Really sick looking when you kick your back leg out a little to tweak it.
Hope that helps where it originally came from