Hiit training can use any type of exercise that allows quick change of pace.

The routine in Tabata’s original HIIT work was just cycling.

Crossfit afaik is a more complex affair than just cycling.

So is the risk of injury due to specific movement or the interval quality of the movement?

I don't think this is helpful. Crossfit appears to prioritize speed and # of reps above all else, and is notorious for people doing difficult exercises with terrible form to try to hit those targets. HIIT certainly emphasises intensity of effort and prefers relatively short rest periods between sets, but not at the expense of proper form. eg when I think of cross-fit weight training or callisthenics I think of lots of snatching and jerking, with HIIT I think of moving slowly with form and trying to hold tension at the apex of the movement.

Don't conflate hiit with crossfit. Crossfit is just one (although well known) type of hiit. There are many other type of hiit activities without the training style of crossfit.