Thanks for the video ATC, the kick before the one you mention is actually an example of what we'd consider a jumping cut kick. If you look at the screengrab below it shows his body is facing forwards and his knee is mostly over rather than fully over as a side kick would be.
It's kind of like the lovechild of a lazy (in technical accuracy rather than speed) side kick and a traditional (knee to chest) push kick.
In the one you mention the hips go fully over but that's because he's trying to put a lot in to it to take his opponent off balance (we normally use a cut kick either as a feint or as a setup to something else).