This might help you understand whats going on, i've taken your pic from your first post and modified it slightly
The 2nd lug, its whole purpose in life is a tie point for a diode on games that have a lamp matrix type setup, so it has NO connection at all to the barrel portion where the lamp fits in, hence why your lamps dont work! Notice in the pic i have indicated on the drawing of the socket(s) where the wires would go.. notice the green/red lug goes directly to the barrel portion whereas yours is insulated.
You would really have to solder the green/red wire to the barrel as i have indicated in the pic below.. this would work BUT.. as pointed out by ballywannabe it might need to be insulated from the coin door where it screws on.
