I have a similar voltage gauge and just because it is digital to two places, I do not consider it quite accurate.
12v on a cold morning is too low, original battery?
When I plug in my voltmeter and ride down the road ('07 chevy), it reads 12.5v to 14.7v as the RVC does it thing. It cycles ever 30min or so after it has charged the battery from being used to start the vehicle. It has a mind of its own. Seems to provide no alternator current at times.
http://www.archivedsites.com/TECH_CONNECT_new/English/2005/TechConnect03%20MayJune05%20Final.pdfUse tow/haul mode to override or turn on the headlights! The battery current sensor will tell the computer to up the voltage.