Right, but it's not that simple. They are called 12 volt batteries because they have 6 cells that operate at close to 2 volts each for to explain the 12 volt nomenclature used. Are they exactly 12 volts - no. The chemistry used determines the voltage seen, and the chemistry used changes slightly as the years go by. The trend is upwards in the last 30 years but only by a few tenths of a volt.
I have no clue as to which or what Cigar-Lighter Adaptor you are referencing. In general terms though, the 13.8 volt level would be referring to the engine running, battery charging voltage levels which vary quite a bit depending mainly on alternator temperature, from
14.75 to 13.3 volts. It is supposed to vary according to battery temperature but doesn't just for an example of the not so perfect real world.Your questions deal mainly with semantics and absorbed double talk, don't get your feet fouled up in that stuff as it's pretty nasty stuff to normally be avoided.