What does the term "Speed" mean in a speaker?

It usually refers to the ability of an attack (the first arrival) to startle you, such as with a kick drum or cymbal. It is also about the detail which can reach you at your seat.

In my mind, this is an artifact of tone, not actual driver speed, but since panel and cones couple so differently to a room, panels are often described as fast, while cones as slow. This is not really what's going on, but whatever. :)

How accurately a speaker system/driver follows a waveform. ie: Fast rise-time / square wave tracing, without ring, overshoot / etc. https://www.sweetwater.com/insync/square-wave-response/

I rarely hear the square wave definition used, but sure, some do.

The square wave has more to do with time/phase accuracy, and something that can be corrected for via DSP.

It is certainly a very rare thing to have a speaker that can replicate it with any sort of accuracy. :)