Neat! The optical term for that effect is a '
caustic (video)', essentially the glass acts like a lens that instead of having a focal
point has a focal
curve. When used for abstract visual effect, it's generally referred to as "lumia", after the work by Thomas Wilfred starting in the 1920s. He was of course using incandescent light sources and colored filters:
https://americanart.si.edu/videos/thomas-wilfreds-lumia-unit-86-154416. In the 70s it became a key part of a lot of planetarium laser shows, taking advantage of the vibrant colors that Wilfred never had access to. There are still some dedicated folks building new and unique lumia generators for lasers:
https://www.youtube.com/watch?v=uJrP585lT4A. Of course now you can also buy all kinds of cheap mood lights from the usual websites that use the LEDs and rotating lenses.
Very low speed DC gearmotors would be the default actuator for this sort of thing, like 10RPM or even less, although the ideal speed depends on the optical components you're using and what kind of effect you want. I know a couple people who have seen good results with geared stepper motors for extremely low speed effects, although they require drivers. Most people move the lens rather than the light source, but either way works. You can also add a moving component between the fixed light source and the lens, which is a common technique for creating more intricate effects. Honestly a lot of this really comes down to playing around with variables until you get something that looks interesting.