A float charger will bring back a battery that's usually deemed "dead" or weak; I have done this a few times before using one of my float chargers. In the state of your battery, you can charge it any of the two ways you mentioned, but the float charger is a plug-in-and-forget type of device; disconnect it when you're ready to start the car. I trust my float chargers enough to leave them connected to my cars during a winter hibernation period which may last up to 7 months at a time.
Trickle charger is an older technology, I believe. Float chargers are usually touted as "microprocessor controlled". I have no idea if float charging can be performed automatically without some kind of an electronic logic controller. When a battery voltage is deemed too weak, my float chargers will recondition the battery by charging it to 16V but they don't linger there; once the target has been reached, the battery voltage is then allowed to settle down to 13.2V when the floating mode begins. Whenever the voltage falls below 13.2V, the charger will bump it to 13.3V then stop and repeat.
One of my float chargers has a charging protocol or programming for an AGM battery. I'm unsure how that compared to my others, but they have been used on a few AGM batteries without any degrading effect.