10.5 volts should make lights work, barely, so 12 is plenty, but yes, volts and amps are different. It could be a dead battery that just does not have enough amperage to do anything. Death could come from not holding voltage or from not having enough amps.
Take to auto parts store for the free charge n check.
You likely damaged it over the winter, most likely you did. If you let it sit all winter not being charged, then you might as well just buy a new one. Some "chargers" will let a battery die also, they just do not give the battery the right voltage and cannot regulate the voltage according the the State of Charge the batteries are at. Over-charging will do damage also.
Get a real deep cycle battery, not these marine ones. They are called Golf Cart batteries typically and are about the same price.
Cleaning the terminals is huge huge too. Sometimes the battery cable under the cable coating can get corroded and green and not conduct the electricity too, but this is unlikely in your situation.
***edit*
I reread your OP.
"12.5 VDC at battery and 13.75 VDC on wire leads from trailer"
after a day if the batter will not get past 13 (assuming you have a modern and decent charger) then the battery just will not take a charge and is dead. If you leave batt unplugged for a day then I bet voltage drops even more. The 12.5 yo uare seeing might just be a "surface" charge.