[masraum]

Battery voltage fully charged should be 12.6 (may be higher due to surface charge).

charging while running should be 13.8 and up (I'm sure there's a max, but can't remember if that's 15, 16, 18, etc...)

Battery likely dead, not alternator.

Batteries can be discharged (<12.5V) due to having something left on (lights, etc...). A discharged battery (if that's the only problem) can be charged back up and be fine.

Batteries can be "dead" not sure if there's another/better term, where they have a full 12.6V charge, but have lost the cranking amps to start the car. I've seen this 100 times, get in the car, start it and drive it and everything is fine. Stop the car, come back out to start the car and you may or may not get a click or even interior lights. Car likely showed 12.6V (until you tried to start it which may have run it down). That battery is just done, no fixing it with a charger.

That probably what you ran into. I've heard that sort of battery death is due to the coating on the plates flaking off or otherwise changing. I've heard of "internal shorts" too, but I'm not sure exactly what that is referring to.