Next to the coil inside the relay frame

Yes, picture is correct.

I recently had a burnt coil (A-9738) from a Gottlieb that had the washer on the inside with the coil. So I replaced it with a used coil (same id) and found that the points were all out of alignment. So I measured both coils and found that the replacement was longer by the thickness of the washer. Once I removed the washer all was good again. Could this be why he has this situation? Anyone else ever had this happen?

I understand the brass washer is located between the relay carbon steel core and the relay frame also carbon steel. Carbon steel can be magnetized brass not. That is the reason it needs to be there.

The burnt coil was nothing to do with the washer, best to put it back and fix the switch leaf points.

Actually thinking back, the longer coil combined with the washer made it so that the coil was hard against the armature so that it couldn't even be retracted. Without the washer everything lined up and works just fine. I really did not have any other option other than to find a shorter coil. I am curios though, at what point did they change the parameters of the coil length? I have no idea out of which machine the replacement coil came from or how long Gottlieb produced coils with the same identification code. Maybe there is some old tech bulletin on the topic.

If it was a newer coil with molded plastic over the end of the core, you should omit the washer. The plastic serves to insulate the coil from the frame, just as the brass washer does for the earlier style coil.

Also remember that those coil hold screws must be brass, too. I always check Gottlieb EMs and occasionally find ones that have been replaced with carbon steel screws -- a no-no -- they can magnetize the frame. Because brass is softer than steel, the slots deform more easily when tightening, and some fixers think a steel replacement will work just fine.