I finally broke down and went against my logic considering the symptoms and boutht a new battery. Note that the Thinkpad has sat untouched for about a week.
I plugged the new battery in and pushed the power button, and the T60 booted right up! I was convinced. So I shut down the laptop, got out the charger, and plugged it in, pushed the power button, and.... the old problem is returned. I speculated that sitting for a week with no battery, no charger, the residual charges within some circuit bled off naturally, allowing the boot-up on the battery alone. The charger I have is a new one, though made in China. A mystery wrapped up in an enigma? Thoughts? I know I must have another charger buried around in my "stuff" somewhere, maybe that's the right fix.