A bad battery will read just like you found. A charger will try to charge it and when you take the charger off, the battery will read 12 Volts and you think it is OK. Put a headlight on the battery as a load. I use an old connector and some clips. Then read the battery with the light on. Less than 12 Volts and the battery is toast. Old hot rod set up for Bright/ Dim was to run both as bright, and dim alone. Saved money on the switch. More money for go fast stuff. Light bulbs were a little short lived but who cared. However, when the generator was not generating those lights got dim fast.
