All are correct. I can't think of any difference. It's mostly a question of how your family did it or what they called it or what the egg coloring kit in the supermarket says. Certainly they are "colored."
Technically a "dye" is a chemical that actually colors the eggshell itself. A "paint" is a liquid that tries into a thin layer of color on top of the egg. The vinegary-smelling kits you can get in the supermarkets are dyes.I thought the fabulous traditional Ukrainian eggs were painted but according to Wikipedia they are in fact dyed.