If the wire colors have faded you can always try to find the other end(s) of the wire for confirmation that you've identified the right wire. If you think you have the right wire on a switch, find the other end of the wire on some other device (as shown on the schematic) and see if there's little or no resistance between the two ends. That should tell you whether or not you've got the right wire.
The Game relay is an interlock relay with two coils that pull the armature and switches back and forth between two positions. Once one coil has fired briefly and pulled the switches one way the relay can mechanically hold its position until the other coil fires briefly to pull it to the other position. It can hold either position indefinitely without power even after the game is turned off.
Lucky Ace Game relay (resized).jpg
The Game relay in this game should reset or "Latch" at the start of a new game when the Coin relay fires and the Ball Count unit is at Zero. The Total Play Meter and Credit Unit subtract solenoid are wired in parallel with the Game relay Latch coil and should fire at the same time.
Shortly after the game starts the Ball Index relay fires when the first points are scored. The Ball Index relay will in turn fire the Game relay "Trip" coil to indicate that the game has started.
The schematic is drawn with interlock relays in their Latched positions. The schematic in reply #2 shows the Game relay switch open which means that the game has started but points haven't been scored yet. Once points have been scored that switch should close which would enable the replay button. Since this is a one player game you want to disable the replay button just after the game starts but enable it again once the game is in progress. Once the game ends the Game relay should still be in the Tripped position, even after the game is shut off, enabling the replay button.