I would call that one the Coin Lockout Relay. There are no switches on that relay, it is connected to a metal bar that goes across over to the coin units. When it is not energized that bar will be in such a position that coins will fall through and come back out the coin return. Thus, in a paying environment if the power is off and someone walks up and drops a coin in, it will fall right back out and they won't lose it. When it is energized, it pulls that bar over and then the coins will be taken and give a credit.

Since that one relay is on all the time that power is on, it gets toasty and can burn up. If it is not energized, you get the exact symptom your describe. I think on most home machines it is disconnected and kept off. You can then manually slide that bar over to the position to allow the coins to be accepted.

EDIT - beaten by Howard

That would make sense. I only have the 2 Bally EMs of similar vintage, Bon Voyage and Monte Carlo. They probably figured it was unlikely that the credit meter would ever get maxed out at 25. Looking at those schematics, there is no other logic for the lockout coil, it is always on when the machine is on (thus accepting coins).

EXCEPT:

It is in series with a score motor switch 1B, which means it will be briefly de-energized while the score motor is turning. My presumption is that this is done so it will return a coin that has been shoved in during the time the machine is working on the reset procedure because that one would not be able to register a credit during that time.