If you're controlling a quadcopter, one limiting factor is how fast the rotors can accelerate/decelerate.

That is, the derivative of rotor thrust.

That is, the derivative of angular acceleration of the vehicle.

That is, the third derivative of the angle of the vehicle.

That is, the third derivative of the horizontal thrust of the vehicle.

That is, the third derivative of the horizontal acceleration of the vehicle.

That is, the 5th derivative of position.

Cannot speak about motion, but in surface continuity you will also look at higher "derivatives" if you want really really smooth transfers between two curves. So you make the transition of the curvature comb of a curve's curvature comb tangetial or so. There they just call the transitions g0, g1, g2, g3, g4 and so on.

I cannot judge whether snap, crackle and pop are things people actually use when they talk about those derivatives in motion.

Exactly, in the end you want smooth motion because of the resulting forces.

But since F = m×a and mass is typically something you cannot dynamically change on the fly, acceleration must do.