I would say during charging you should see 13.8 - 14.4 volts. What you see is a little on the low side but not desperate.
What I would do is to charge the battery - using a mains charger, for example, so that you know it is fully charged. Let it rest for a few hours and then measure the terminal voltage, which should be over 12.5, and then switch on, say, the headlights and watch the voltage over a period of a few minutes. The voltage will drop a little but if it drops to 12 volts and below the battery has had it.
If the battery appears OK it's time to check for something draining it but, as you say, if it's dropping to 11.8v overnight it would be flat after another day standing.
The ABS is probably seeing low supply voltage as it does a self test when vehicle motion is first detected. Giving a little time for the battery voltage to get dragged up by the alternator is probably helping here.
Kevin