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.

All are correct. Paint means you use a brush, dye and color means you dip the eggs into colored liquid. I just did this with my family yesterday!