The battery meter looks ok to me, mine always hovered around 12-13v. The dash one is battery voltage not charge voltage.
The rac tester could be at fault, if the alternator wasn't working the battery wouldn't be charged, also engine speed would drop as more electrical items are switched on, until it eventually cuts out.
One of these is what i used,
http://www.ebay.co.uk/itm/Car-Battery-Alternator-Tester-6v-12v-6-Volt-12-Volt-High-Load-/201081182614?pt=UK_Diagnostic_Tools_Equipment&hash=item2ed15f5996
If you need a replacement alternator let me know.