Would a sulfated lead-acid battery charge at 2.1V per cell?
13V read on your multimeter is probably RMS value. There are probably relatively high voltage pulses.

I would just charge that battery and try to use it. If it does not hold charge - you should replace it.

I was experimenting a lot with desulphation and... for car battery it does not make sense. If there are big sulphur crystals - you have to use chemical method (instead of electrical method with fancy charger). If there is just small sulphur dust on electrodes - desulphating will not help a lot. It is waste of time.