Star Wars, GOT, and Ghostbusters are recent Stern machines with video modes...
The first video mode is in Mr./Ms. Pac-man. 1983.
Quite a few machines have 3 video modes:
Williams Indiana Jones. It has 3 video modes.
Also, Black Rose has 3 video modes but none of them are particularly well done IMHO.
Junkyard has a similar progression of two separate video modes which get increasingly difficult. And, it also has 3 video modes (Time Machine video saucer mode).
Doctor Who is worth noting because it's actually very well done.
My favorite modern video mode is BSD but nothing tops Mr/Ms Pac-man. No one even understand how to play it or how complex and fun it really is to play. I've complete 3+ mazes on it- it's very difficult to get that far in it.