I'd wager you have a Delco 10SI "3 wire" alternator and that the wire from the door bell button goes to the #1 pin on the edge of the alternator. The alternator requires a momentary voltage "kick" to start it charging. There are several ways to accomplish this. On yours you start the engine then press the door bell button to start the alternator charging.
This same function is usually accomplished by a diode or "idiot light" that is automatic.