I can’t speak to pinball plastics in particular but we do a lot of printing on acrylic, PETG, and other plastics for my job.
Ink is put down in layers on one side of the plastic, so the order of application all depends on what finish you want to have. You can hopefully see in the photos of the cat below how the side with the ink is more of a matte finish.
So then it also depends on what type of image you want the end result to be. Sometimes we want some transparency so we don’t use a white layer. Other times, like in this example, we want the colors to be really bright and pop, so we need the white behind the artwork as a base. Finally, if we want the art on both sides, like this, we do COLOR>WHITE>COLOR. A one sided matte finish might be WHITE>COLOR, while a glossy one sided like on most pins would be COLOR>WHITE.
I’d have to confirm with our vendor, but most of our printing now is digital, not screened. We print first and then do the cutting, and in this case, the heat-bending.694476AA-EEE3-4369-A9B5-E598F7EB881B (resized).jpegDC583C52-4D5C-45E5-B8D3-58C812B052C2 (resized).jpeg42BEF5E5-BE84-4336-A4B9-401F53C5C49C (resized).jpegDCE4A275-A171-4701-B8E0-D4A6297B3FF8 (resized).jpeg