Electrons do move inside a battery as a result of a chemical reaction inside it. The electrons move towards an electrode that conducts them to a terminal that can be connected to a wire or fixture.
Powerful chemical reactions occurring inside a battery form ions and release electrons. These may result from the interaction of lead with sulfuric acid, or lithium with cobalt oxide.
These are both re-chargeable batteries because the chemical reaction can go forward to produce electrons, or it may be reversed, when the battery is being charged.
The potential energy of a battery is dependent on the number of cells present, the chemicals used in the reaction, how much it is charged and its age.
A second terminal is required on the battery to allow electrons to return to the battery from the device to complete the circuit.