I think they only go into a full charge mode if voltage drops below 80%. 80%-100% the green light should be blinking though, as it's in a lighter charge mode maybe? Then when it hits 100% it goes into float mode.
Resting fully charged 12-volt flooded lead acid batteries are around 12.8-12.9 volts (some AGM are a bit higher, Gel are a bit lower). 12.0 volts is considered "dead", so 12.4 volts on a resting battery means it's about 50% charged.
Some chargers will cut themselves off if they have run for an extended time without bringing the battery up to a full state of charge. This is a safety mechanism so they don't keep trying to pump amps into a damaged battery. Even some "dumb" chargers have a timer built in. Some chargers can detect a load on the battery and warn you about that. A couple of my chargers will test for and detect a bad cell.
If you fully charge a flooded lead acid battery, let it sit overnight (with no load connected), and find that the voltage has dropped significantly below the 12.8 volt fully-charged resting state, it's likely you have a shorted cell and/or are heavily sulfated battery.
a couple of battery diagnostic tips:
If you let the battery rest for a day or two and it drops to 10.5 volts or so a cell is likely dead.
If the battery won't charge to more than 12.5 volts or so (assuming you have a decent charger connected) it's likely that the battery is sulfated. Note that some battery chargers will still show the battery as "Full" because the battery has stopped accepting charge. Connecting to a battery charger that has a desulfation feature
might bring it back, though it can sometimes take quite some time to do so (weeks, if it is heavily sulfated).
