You said the the car died BOTH times with battery disconnected when you turn on headlights. Sounds like alternator is putting out enough charge on it's own to power ignition, but when loaded down with big load like headlights, it isn't... Does this vehicle have the alternator bypass wire added, so that the main current does not go through the firewall connector?
If not, then there could be enough resistance in firewall connector to drop voltage coming into keyswitch and going out to ignition enough to keep it from firing, especially when idling (lowest alternator output), and battery disconnected (no backup 12 volt supply).
Check what alternator puts out at idle... should be greater than 12 volts, and when above idle should be 14-15 volts.
Sounds like you are getting down to like 8 volts with headlights on... ignition no likey that low...