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Ive been powering the Xbox on, I found that if you wait for the third reboot(where it will then turn to green and red flashing) then connect the cable it doesnt frag(I assume this is power off right?).
From what I have read, it is my impression that xbox 1.0 to 1.5 behave different than 1.6 and 1.6b
The older models left the power on when the eject was flashing red and green. The 1.6 models turn the power off. An indication for that is the fan on the rear of the xbox. On the older models, that fan keeps running, on the 1.6 it is switched off as well.
I wouldn't test with a separate supply. The eeprom is not the only chip connected to the supply rail. If you power it with another supply, you might damage some of the other chips. The eeprom is fine between 3V3 and 6V, but I am unsure about the others. Some might be designed for 3V3 only with a 10% toleration.
If you unplug the pc, are you able to boot your xbox into an error screen with the reader connected, or does it frag? In my opinion, it should boot as it should without the reader connected.
regards.