Another electrical system question

[Frodo]

This is an offshoot from my “Battery choice” thread.

Having been pretty convinced my battery was dead or dying, I had disconnected it from the cables in the car. (I was curious what it weighed.) I decided to charge it on a trickle charger for a few hours. A few hours after finishing that, it dawned on me that I had this analog multimeter, so I dug it out and checked the battery. It read better than 12V. Decided to put it back into the car, which started up nicely afterwards. (A couple days ago, I was left stranded away from home when the engine---having started an hour earlier---wouldn’t crank.) Went for a drive, pretty much everything was fine. Got back, shut her down, checked the voltage again. Again, it was over 12V. The question: the part that was not “fine” on my drive was the fact that the battery light came on (and stayed on).

Suggestions as to why? Is it probably just due to a bad battery, or do I likely have something more serious, like an alternator problem? Or is it impossible to speculate intelligently without additional diagnostics?

Thanks for any ideas.