What? If you have an AGM (absorbed glass matt) battery then the only kind of charger you should ever be using is one designed for an AGM battery.
Flooded cell batteries use a standard car charger
Jell cell batteries have a specific charger for them as well.
Each type of battery charges differently some slow and low some fast and hard like an AGM using the wrong charger will shorten the life and lessen the performance of any battery.
As for resting voltages 12.7 is a little low but how old is the battery? What is the battery used for or is it just sitting around not being used?
If it's cold out the battery will rest lower...my datasafe rests at 13v when it's 70+ degrees but when its cold out it rests at 12.0 - 12.2. Car still starts up with no problems