When are you going to measure the voltage? If you measure it while its charging it'll be higher momentarily, as high as 14 volts then drop to 13 in time after its done charging, so depending on what kind of load it is you may not want to turn on until its full, or maybe it doesn't matter idk, but if its charging it'll probably say 12.7 and if it were to be disconnected it would drop lower which kinda gives a false sense of it being done
Yeah can't hurt to shut off earlier, although that kind of battery can probably handle it,