Normal battery behavior is to show a higher than normal voltage reading after charging until the "surface charge" is bled off. Turn your headlights on for a minute, then measure the voltage at the battery (headlights still on). You should then see 12.6 +/- .2 or .3 volts.
A defective battery with a shorted cell will not allow the charging voltage of 13.8 to 14.4 volts to be measured. It would measure about 1.6 volts lower than expected for each shorted cell.
Just went through this thhis week. Battery with a shorted cell combined with an alternator failure. Spent all day yesterday replacing the alternator on my back on a gravel driveway. No fun..
Your alternator is toast, just quit dicking around and replace it.
Steve