Laszlo.
So this is what Ive observed happening with this circuit.

the ECM puts out 12 volts when the key is first turned on. Then the engine starts and it continues putting 12 volts out thru the inertia switch and into the relay.

When I bump the switch (and it closes), the engine stalls within a second. When that happens, the ECM wire loses all voltage. Since it's connected to the C terminal on the switch, and the switch mechanically transfers the connection from the NC wire over to the NO wire...that NO wire now acts as a ground to the light/buzzer.

I first tried connecting the light/buzzer the "other way" (thinking the NO wire would now transfer the 12 volts coming from the ECM wire to the C wire within the switch, but since the engine dies that NO wire loses all voltage...so nothing powered up the light/buzzer with it connected in this manner).

So the NO wire becomes a ground when the engine dies, the ECM quits putting out 12volts, and that allows the light/buzzer to turn on (because of the 12 volt connection to battery power)

probably lost y'all on this by now.