Nope, your painter is incorrect, the interior parts of the door that you are referring-to are painted the interior color.

Look at the exterior door skin and at any outer edge of the door. You will see that the outer skin is bent-over towards the inside (on all outer edges of the door). There is 1/2" or less of a "lap" where the outside metal overlaps the inside metal. The outside color runs to end of the outer-metal overlap.

I will see if I can post a photo tomorrow.