That's not how my '74 Duster worked. If the key was in the ignition and the door was open, the buzzer buzzed. Running or not. Not sure why yours would be different, pretty sure they both use the same buzzer.
Only way I can think of this working is if the key-in switch is somehow disconnected when the ignition switch is turned. Maybe the lock cylinder is shaped differently and doesn't depress the switch when turned? After who knows how many changes to my car, no idea if I just ended up with wrong parts or something. It would make sense that it should do that, I just didn't see a way to make it happen without some wiring changes.
It was one of those things that bugged me enough that while I was doing some other rewiring, I added a buzzer from a Grand Cherokee. I used a wire from the seatbelt interlock so I had one that was hot when the ignition is on to keep the buzzer quiet.
I wanted the buzzer when the key was in the ignition and the door open, but I don't need it to tell me that when the motor is running. So I didn't want to remove it all together, just upgrade it. Plus, I added a headlight warning to the circuit using the same buzzer.
