Won't start unless I take out, reinsert battery

Lattitude D410, 2.13GHz (board G8338) 2gb RAM, 80GB HDD, Windows XP Pro 

SITUATION: A weird one. I invite any and all geniuses to comment.

Laptop no boot unless I remove the battery and AC adapter, then reinsert battery.

Upon reinserting, it powers on by itself. Don't have to touch power button. If I do not reinsert battery, it doesn't start on AC power. 

This is apparently a fairly common problem - unfortunately all search results take me to forums that are either a) ad- or product-driven b) verbose c) confusing d) ill-informed

Anyone know why, how come, how to fix it?

CLUES:

Problem began with laptop simply shutting down in a very weird way. It would just -- snap-- go black, instantly and seemingly randomly, no effort by Windows to shut itself down. So I'll just go snoop in my Events logs and get the answer, right? Nothing!!! Neither Application nor System nor Security events log has anything to say about anything  unusual happening in the seconds prior to the shutdown.

Well, fine, Windows is no help. But my good ol' Dell Diagnostics Utility will tell me, right? No! It found the computer to be in a state of pristine.health It passed every test both short-form, long-form, custom -- not a single hardware problem seen.

I somehow fixed the shutting down randomly problem, but still have the need to remove battery and AC charger, then reinsert battery to start.

WHAT I"VE DONE/RULED OUT:

• Put the HDD in another computer and did a Windows checkdisk. It completed all 5 phases.
• Beside Dell Diagnostics, I did extensive third-party hard drive and memory tests. Everything passes.
• Ran a SFC scannow with my install CD.
• Hit my dog
• Entered recovery console and did a bootcfg/rebuild.
• Tested with a second identical working battery
• Hit my wife
• Tested with a second identical working AC adapter
• Took out each of the 2 memory chips, one by one, to see if either was causing the problem
• Got hit in the head by my wife
• Dog bit me 

Strangely, none of these worked. I'm afraid I have lost my touch. Please, enlighten me, someone

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