That's not really accurate since the "electrolyte" changes phase requiring external storage (H2O -> gaseous H2) and the re-oxygenation step cannot be performed

That's not really accurate since the "electrolyte" changes phase requiring external storage (H2O -> gaseous H2) and the re-oxygenation step cannot be performed (yet) in the same device as the electrolysis. It's not an electrochemical battery but I can agree with seeing it in a general sense as the an energy storage device, in the same way a dam is a "gravitational battery".