Isn't that the acceleration? With constant acceleration (thus zero jerk) inertia makes your coffee move inside the cup – and spill, if you don't pay enough attention.
1) coffee is always under constant acceleration (g).
2) constant acceleration (say in the X direction instead of Y) would just mean a constant "tilt" within the cup. compare this to an airplane that is not "accelerating" and just at a constant X velocity, coffee would look "still/flat" in your cup.
3) its only the jerk that changes the "tilt" within the cup (and hence causes the spill)
Constant acceleration will cause the surface of your coffee to tilt, but not spill it unless the cup is almost full. Jerky motion (non-constant acceleration) causes it to slosh around and can create resonances that disturb the fluid level far more than a constant acceleration of the same magnitude would.
If you move in a circle around a corner, you have constant acceleration, otherwise you would go straight (centripetal force needs acceleration to exist). yet it is possible to move a coffee cup in a corner, as long as you tilt it a little bit. So acceleration is not the issue
However if you suddenly change the direction of the coffee cup, you introduce jerk, because you accelerate the acceleration (you change the size of the circle means you change the acceleration, therefore you introduce jerk = coffee spilled on the floor)
