I'm glad you liked the video and seem to have a good idea what the motion actually is.
I like you came up with your own terminology - "side movement", "rotation".
Interestingly Jim McLennan calls this movement "forearm rotation" when speaking about it in the serve.
The Pete Sampras Snap http://www.essentialtennisinstructio...-sampras-snap/
Probably the most common term you will hear for this term is "pronation
Bruce Elliot and Brian Gordon, Ph.D. did the pioneering work to show that internal rotation at the shoulder
is responsible for the whole arm rotating - it is not just the wrist or forearm - and the main forces that cause that whole arm rotation come from the shoulder area.
Obviously as the whole arm rotates, the forearm and wrist does so as well.
This probably explains why some emphasize the feeling that it is the wrist that is "snapping" around, while others feel the forearm rotation is dominant - but the whole arm is rotating.