I put the battery is charged, and Duolingo accepted it, but noted that, The battery is full is another correct translation.
I agree that The battery is charged (or The battery is fully charged) are the best translations, at least here in the US. So, for example, I would say, "I packed the camera, and the battery is charged."
der Akku is an accumulator, a rechargeable battery.
die Zelle is an (electrical) cell.
die Batterie is, strictly speaking, a battery: a group of individual cells working together.
However, in colloquial language, you may hear Batterie even for an individual Zelle and also for rechargeable ones.