How does that battery/production distinction work? How is it a battery if it is consuming/burning a fuel? More as a stable fallback or something?
and then put back energy in the spent fuel to make new fuel again
then you have really a battery. That's how li-ion batteries work. The issue is the efficiency: how much of the energy you used to recharge the "battery" (iron) is going to be available when you discharge (burn) it
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).