That web page has a very through description of what the circuit does and how it works. It's more complicated because it sequences two control lines and takes great pains to make sure things are done in the right order with the right timings.
You haven't said much about how your car PC is powered, so it's impossible to assess whether this circuit will work for you. When the engine is off will the PC be powered but in standby mode, or will it be powered off, or ...

Is it powered from the car's battery or does it have its own power source? The more information you can give the more help you can get.