Your 12.5 V test, was that after the battery was sitting for a few hours not doing anything?

12.66 for a lead acid battery is full charge. 11.70 complete discharge.

As for wet stacking, in the old days starters and batteries were not so good. Leaving them run was needed. If dealing with sub zero temps and no where to plug in over night, then idling is the only option. However, batteries and starters are better now making shut off possible.

Big trucks no longer let idle as this is hard on fuel for the amount of energy that comes off the motor. Large trucks now have a small diesel engine running a generator / AC compressor to provide energy when stopped. It has been illegal in many states to idle a over 5 ton truck more than 5 min for some time now.

Wet stacking occurs due to incomplete combustion ( cold cylinders ) at light loads. This allows carbon to build up in the combustion chamber, injector tips, exhaust valves, manifolds and muffler. You will also get fuel dilution and reduced engine life.

Guys that work on diesel back up generators / portable welders know all about wet stacking as these engines are frequently lightly loaded for extended periods.