Non-rechargeable batteries are also consuming a fuel. The products just stay in the same enclosed container. Same with rechargable batteries, just that there the process is easily reversible.
It is a bit of a fuzzy distinction. Batteries are typically simple chemical reactions that cause electrons to move around. But viewed from the outside a hydrogen fuel cell behaves the same; so why not call this one a battery too (especially since the process is reversible).
A useful distinction seems to be that batteries are solid state and don't use high process heat, else something would be seriously wrong. Of course, the underlying reactions are probably very similar if you look at them with a chemist's eye.
It is a bit of a fuzzy distinction. Batteries are typically simple chemical reactions that cause electrons to move around. But viewed from the outside a hydrogen fuel cell behaves the same; so why not call this one a battery too (especially since the process is reversible).