Why do we say that batteries/energy storage is something that stores and releases energy?
A device that is able to convert some amount of heat into zero-entropy energy is ~just as useful as a battery of the same capacity. It's obvious that such a device can exist (a thermos with some amount of heat capacity inside at a temperature lower than environment + a heat engine is such a device), and it has at least one significant advantage over a battery: it is not necessarily able to release a significant quantity of energy on failure/destruction.
One can easily create such batteries in that exact way (by using a thermos and heat engine). My rough upper bound is that the highest "energy" density one could get without phase transition would be ~2MJ/kg[*] (by using gaseous hydrogen as the medium), which is a bit better than current batteries (but is an upper bound that ignores any practicalities of handling hydrogen, inefficiencies other than thermodynamically necessary, and the weight of infrastructure). I've looked at a few phase transitions (e.g. ice/water or liquid/gaseous nitrogen) and they don't seem to be able to give anything even close to the value for hydrogen (or, for that matter, mere Li-Ion batteries).
Are there other ways to store negative entropy? I imagine that chemical ones "should" exist, but I have terrible intuition for entropy changes across chemical reactions to even know where to start looking.
[*] I've taken half of the energy that would be needed to heat hydrogen from ~0K to ~room temp (half because efficiency will change linearly between 1 and 0 as the temperature of the hydrogen increases).
tl;dr Are there any better ways to store negative entropy without storing energy than storing very cold hydrogen? Is there some sort of fundamental limitation in play?
