The Battery Light works by current flow.
Any light bulb works this way, if I hook up 12v to one wire on a light and 0volts(ground) to the other wire on the light, the light will come on because current is flowing through the filament.
If I hook up 12volts to each wire on the light it will not light up, because there is no current flowing through, both wires have equal voltage.
When you turn on the key, 12volts should flow from ignition switch to/thru the Battery Light and then to the alternator which is 0 volts since it is not turning and producing voltage, so battery light is on.
When you start the engine a working alternator will produce voltage so Battery Light goes off, equal voltage on both wires.
With alternator running system voltage is now 13.6volts(approx.), so both wires on the Battery Light are 13.6v so no current flow and light stays off.
If alternator starts to fail and it's voltage output drops below 12.5volts(battery voltage) then Battery Light comes on because current is flowing thru the bulb from Batteries 12.5v to alternators less than 12volts.
This Battery Light wire is also the ON/OFF switch for alternator.
An alternator is built much like an electric motor, so it will draw power in some cases if left connected.
So the 12volt/key on turns on the alternator, with key off alternators rotor is disconnected so can't act as an electric motor.