Yeah, the charger detects if the battery is low so it can stop charging when it is no longer needed. That’s why some chargers have a light on them, so you can see (if it is lit) that it is still charging, and if the light is off, the battery is fully charged.
Actually, a voltage of zero indicates a depleted battery and signals the beginning of a charge cycle. In that case, a good charger will go through a recovery procedure prior to charging.
The float voltage is at its highest following a charge, still below the charge voltage. A charger stops when the current drawn by the the reverse-biased battery drops below a threshold.