The high resolution but limited colors example looks like screen printing. Screen printing involves squeezing ink through stencils supported by a fine mesh. It can produce excellent detail, but you need a separate stencil for each color so the number of colors is usually low.

A lot of old arcade and pinball machines used screen printing for the cabinet artwork. I like it better than standard CYMK printing because it can produce more saturated colors.

https://en.wikipedia.org/wiki/Screen_printing