The video doesn't work for me -- says it is private.
Servo implies that there is an encoder on the shaft. Motor can be any kind.
Steppers have distinct coils, so with appropriate driving signal there are discrete stable points along a single rotation - steps. Not the smoothest motion but cheap, easy and can bear a load -- with these you move things into position, then let everything settle down before doing something. And you don't need an encoder because you can keep count of the steps, assuming the mechanism is well made.
If you want to do something while in motion then you probably want a dc motor. Smooth motion but more work and expense typically. Harder to hold still under load. Needs an encoder on it if you want to know position.
I've remembered since my first post that once programmed a multiaxis controller made by a company called Newport that would coordinate along a defined path. It would interpolate as needed, you just defined points and time. You don't need any custom software or hardware -- I bet you could do it yourself if you had the manual.
The harder part might be the mechanics of the device you have in mind.