I'm curious, are there types of battery that don't use ion exchange? I'm familiar with physical energy storage using flywheels or compressed air; I suppose you could call something like that a battery if set up to spin a generator.
That's an interesting question. As you say, it depends on definitions.
I would say that some betavoltaic devices would qualify, and perhaps some phase-change stuff that I can't recall right now. Supercapacitors are essentially batteries but I seem to recall they actually do ion exchange as well.