I just got home and just now saw this thread. I've been out all day today repairing pinball machines, arcade video monitors, and a CD jukebox today.
Your picture shows the back plate sub-assembly of a Coin Acceptors, Inc. (a.k.a. Coinco) model 2802 amusement game coin door. Where is the rest of the door?
Here is the parts list of it:
http://arcarc.xmission.com/PDF_Coin_Mechs_Etc/Coinco/Coinco_2802_Series_Game_Door.pdfAnyway, not everything is hooked up on your door. Someone has it minimally wired for just basic functions.
The black wire is common to the coin switches ("Common" terminal) and the outer shell of the light sockets.
The gray wire goes to the "N.O." (normally open) terminals of both coin switches. This would make both switches to be Coin Switch 1.
I can't see the color clearly, but it looks like the white with long red stripe is going to both of the center contacts of the lamp sockets.
It looks like the white with black banded wire is going to one side of the connector to the counting meter.
It looks like the white white red banded wire is going to the other side of the connector to the counting meter.
What is the operating voltage of the counting meter? It should be stamped on the meter along with the manufacturer's name. Which of the two meter connector wires is going to the banded end of the diode on that connector?