As of this morning (Sunday), the voltage across the battery terminals reads 12.69 volts. Evidently, the battery is fully charged and the charger has tapered off to a minimum.
Yup. A lead-acid battery which is fully charged and has "relaxed" for several hours should have about 12.7V across its terminals. While charging is in progress, the voltage can reach 13.8V (and a bit higher with some chargers).