So what's missing. The controller box can somehow tell whether the car is running, how's it doing this. The first couple of ideas I had were ridiculous and complicated. A piezo sensor to sense the engine vibration, or sensing the voltage droop of the battery when the alternator kicks in, but how does it know the car is off?
After thinking about it a bit more I think I know what is happening. When the car is running, the alternator charges the battery, and when it does this there is 13 to 14 volts across the terminals of the batteries. This must be how the controller knows the car is running. 12 volts isn't enough to turn the system on, it can only be done with the higher charging voltage. I won't be able to test my theory because the unit has been installed, but it seems to make sense. I like the idea, it's simple and elegant, keeps components to a minimum, and makes after market installation easy for someone at home.