A volt meter is not always a good read of the condition of the battery. While a volt meter can be indicative of a bad battery, you, also, can have a volt meter reading 12.6v on a battery that's no good. All the voltmeter does is tell you at what voltage is between ground and the positive post of the battery. Sometimes that's telling of the battery condition and sometimes not. Whlle voltage is a clue, to tell the condition of the battery, you need more than just the voltage. You need a load test.
First thing I would do, considering the age of the car's electrics, is do what "SVO" suggests and validate the car's volt meter using a DMM. Next, I'd have the battery tested with a "battery tester". Most DIYs do not own that piece of equipment, so you may need to go to a service shop to have it done.
If the battery tests ok, next thing I'd do, is run the engine and double check the charge light with the DMM. If the DMM confirms system voltage is dropping below battery voltage at idle, I'd first check alternator wires and connections. If you find no problems there, you can either have the alternator bench tested or just assume it's got a failing regulator or rectifier bridge and replace it.