I am familiar with Rallycross from Norway. We had a really hot headed guy in Norway. He started young, to his mom's worst fears, having the moped engine in thousand pieces in his bed and sleeping with the parts.
That's must something different. Motocross is motorcycle racing - wheel to wheel - on closed dirt course.
Rallycross is a SCCA's way to get back into the closed section rally game, even though they sold of the sanctioning of US 'Pro rally' which was basically an FIA type event like the rest of the world does, but also allowed non-professional entrants.
It's basicly an autocross on non-paved surface.
Big difference is every run counts, and the course may be altered during the event.