A battery is a simple thing but I'm afraid, the concept just doesn't work in my mind. The confusion is basically that whether it stores energy or simply it makes it by converting chemical energy into electrical energy.
A battery is a chemical reaction. A simple cell will produce charge by ion exchange and consume electrode material. A depolarizing chemical in the cell keeps this reaction going. The rate of the chemical reaction is increased by drawing current from it - you alter the equilibrium of the reaction. If you draw current or not the chemical reaction continues and so the cell will have a limited shelf life. Usually the reaction is in one direction only but in rechargeable cells the reaction can be reversed when current is fed into the cell. It doesn't store the charge as such but uses it to reset the chemistry.
Fuel cells consume chemicals (typically hydrogen and oxygen) to produce electricity.
Your question is a false dichotomy. Batteries do store energy. Batteries also convert chemical energy to electrical energy. Chemical energy is the kind of energy that batteries store, and when that chemical energy is used by allowing the reaction to proceed, electrical energy is the result.