Dell Latitude D600 won't boot unless external monitor hooked

Dear Friends, Can anybody please help me with this weird thing?
My laptop: Dell Latitude D600.
After power failure the computer failed to boot. It did beep signaling error, but can't recall the sequence. What I saw is that any time I try to turn on, the boot starts, however the screen is black, but faint picture of the booting sequence can be discerned. Then an orange led on the right led bus turned on, and the computer shuts down itself.
I suspected that the inverter blew out, and purchased compatible one second hand. However, it is important to say that the computer boots and works just as always if external monitor is hooked behind. The monitor can be turned off, and the laptop will still go up without problem. However, if the external monitor cable or power cable are unplugged, the same kind of failure to boot ensues. Also, if the external monitor is hooked and the laptop had been booted, even after any time, once I unplug the monitor cable or its power cable, the orange led turns on, the computer turns off.
When I received the second hand inverter I installed it, although it isn’t identical to the original one. After installation the monitor came back to life, with light and picture, but either during boot, or short after that, the orange led turns on and the computer shuts down. I tried to change BIOS settings, and several times I could make my computer to boot for long period of times, but during boot is always show, for short time, the orange led. Also, new warning came regarding the battery exhaustion and that I need to plug the ac adapter, although the adapter is ALWAYS in since the battery is old.
Now, the laptop never boots, just like right after the inverter blowing. However, in external monitor is hooked, as before, the computer boots in a natural way, no orange led, and both external and laptop screens are working beautifully.
Now, the very best thing for me, is, unsurprisingly, to fix the entire problem and that the laptop will just function regularly. As a worst case, I don’t care if the laptop will beet even without its inherent monitor since I always hook it, eventually, to projector. However, I need it to be immune to external monitor unhooking, and that it will stay on until I hook it to the projector.
Any ideas? Can I alter the BIOS to ignore failure modes? To omit the internal screen? Anything wise at all??
Thank you all in advance,
Yours,
Tournesol

An interesting update: When the laptop is running (with external and its own screens together), and the external monitor is unhooked for a very short time, and glugged in immediatly, the orange led shows up, and a warning that the battery is too low and I should plug in the AC. Note that anyway the laptop is always on ac since the battery is low, and anyway, why it is responding to the external monitor. BTW, if I unplug the monitor, and fail to plug it is fast enough, the computer goes down as if the battery is dead and its not hooked to the AC.
 
Sounds like the motherboard has been damaged in a subtle way and it will have to be replaced. You have two problems. Usually when an external monitor only works normally it indicates the the LCD panel or inverter is bad. You have had some success changing the inverter, but the laptop shouldn't shut down on its own
 
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