I had a similar problem - situation. My laptop would run from a wall wart
supply, or battery, but had no 12 volt input to connect it to a car cigar
lighter socket.

Except we aren't talking about using an inverter to turn 120 VAC into 12 volts
DC to run a cigarette lighter adapter on a laptop.

We are talking about a power supply in a game console that has multiple B+ at
12 VDC, 5 VDC, 3.3 VDC, and 3 VDC as well as multiple grounds which powers the
motherboard, hard drive, and USB interfaces. It happens to use a lot of
industry standard interconnects, but that still is no assurance of perfect
compatibility with parts not designed to be used with an X-Box.

I still recommend getting a proper replacement power supply if nothing else is
at fault in the original poster's X-Box. - Reinhart