A good battery will give 12.6 - 12.8 volts when fully charged, as the battery ages and the interior is eaten away, the acid is used up by the absorption of the lead and other minerals and the electrolyte becomes weaker. The weaker electrolyte is more liable to freeze in cold weather and the battery loses further power, not able to give full power at any time the lights will dim as they are relying on the alternator to provide the power.
A discharged battery will give less than 12 volts, down from around 2.1 volts per cell, 2.1 x 6 = 12.6 V. The electrolyte is re-invigorated by charging but will not have the ability to hold a charge for too long.
Relays may fail to operate or quit operating when the amperage to it drops too low. When the lights dim it is because of low amperage, the volts may be quite high.
Two lantern batteries put out 12 V (but low amps) when coupled together but will not drive your electrical system, it needs sufficient amps.
Thats my somewhat ignorant understanding of batteries, amps and volts.