Non-solid-state batteries have liquid/gel electrolyte.
The term has obviously been selected because of how revolutionary solid-state electronic devices were/are. And it's not like they selected it after a revolutionary energy storage device was introduced. Solid-state batteries (cells) are an incremental improvement, at best.
And because this is the web: I am not being negative about the technology. I happen to be of the radical opinion that incremental improvements are good, actually.
As for wood batteries,
Most usage of the term 'battery' is independent of technology, outside of usually wanting electricity as the interface. The fact that most batteries people talk about happen to be electrochemical cells is incidental -- it's irrelevant to whatever they are actually talking about. So yes, a lot of the time 'battery' happens to refer to electrochemical cells, but it also refers to energy storage devices in general, which wood obviously is.
The problem with wood being a "solid-state battery" is surely the solid-state part.
Q: I mean how does that even work, solid-state combustion?
A: No, it's still obviously a normal chemical cell. You do a chemistry and these special "electrodes" turn the result into electricity.
Q: Can you recharge it?
A: Well, not in your own home, but in your backyard, maybe sort of.
If we are to use the term 'solid-state' to refer to anything that is solid, as in "solid-state battery", and we can get electricity from burning wood, I don't see the problem with referring to wood as a solid-state battery.
Wood isn't an energy storage device. You can't have a piece of wood where the energy has been depleted. There is no device. Wood is the stored energy - it's a fuel. You consume it in some device, like a furnace, to generate energy.
The "wood is a solid-state battery" thing was an absurd rhetorical device I used to point out -- particularly in a field so significantly connected to electronics -- that merely substituting a liquid for a solid does not justify using what will obviously be understood to be a reference to solid-state electronics.
P.S. A coal-fired steam engine is a solid-state motor.
A steam engine isn't a solid-state motor.
For the purist in doctrine and structure, wood is not a battery. If one is somewhere between neutral and radical, then wood is a battery that was built by its integrated disposable solar cells.