This illustrates the spin axes for the kick, slice and flat serves. There is only one spin axis. But any spin axis can be specified by 3 components of spin in a coordinate system. Here they use
1) pure side spin, a vertical component, Z(upward)
2) pure top spin - a horizontal component , Y (leftward)
3) gyrospin - forward component X(forward)
The longer the arrow is, the higher the spin rate. The kick serve is significantly higher in spin rate.
In the OP, what direction is "lateral"? I assumed that it was any direction perpendicular to the ball's trajectory. ?
If you think about the direction of the spin axes indicated by the arrows you can understand the bounces, especially the bounce to the right for the kick serve.
Imagine the spin axis for the kick serve and the ball spinning. Lower that spinning ball on the court and the first felt that is touched by the court will come out of the page to the left side. That felt motion would make the ball bounce to the right.
The most important feature for the kick serve and bounce to the right is that the spin axis must be tilted more.
Consider also the felt motion for the slice and flat serves - first felt touch is nearer the axis = lower felt velocity.
The racket path is complicated for a high level serve and videos show racket head rotations in 3 D and impact effects. But the net effect of impact has to be as shown by the ball spin diagram. The racket
more or less has to move perpendicular to the spin axis shown in the picture. I consider that mostly 'lateral' to the ball's trajectory and it is required to show diagrams or measurements because words cannot describe tennis strokes accurately.
From earlier post -
"The head of your racket has no upward motions when it is contacting the ball. The racket has to rise in frames before, during and after impact.
Single frame on Vimeo hold down the SHIFT KEY and use the ARROW KEYS."
Does anyone agree or disagree?
Discussion of the diagram.