Based on a reading of the article, apparently li-ion batteries do use liquid electrolytes to move the lithium ions around.

No, solid-state usually refers to a component that undergoes no mechanical or chemical change while in normal operation; conventional batteries undergo redox reactions while charging and discharging.

I'm not an expert, but the article mentions current LiIon using "liquid electrolytes" a few times.

I found this [0] when trying to answer the same question myself

[0] https://en.wikipedia.org/wiki/Solid-state_battery

You only have 3 choices, solid, liquid, or gas.

If a liquid electrolyte is used, it would be a liquid-state device, not unlike a lead-acid cell or aluminum electrolytic capacitor.