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. :)

Speed is yet another quality that you may or may not want to prioritize in what you look for in a speaker.  
The Spendor D7 for example are a fast, nimble speaker.  They have a good jump factor and respond quickly to slight changes in rhythm, a lighter more airy quality to the sound.  They are attention grabbers and less relaxing for example. It isn't so obvious in all music though and for me is not a quality I necessarily prioritize.  

@eric- Certainly, much of what’s causing phase/freq anomalies, etc, in a speaker system, can be corrected with DSP. ie: https://www.audioxpress.com/article/a-loudspeaker-that-can-play-square-waves Far as individual drivers(or- identical multiples) and reproducing square waves; my thoughts on correction, would be more along the line of what Infinity and Genesis(et al) pursued, with their servo-contolled stuff: https://www.psaudio.com/pauls-posts/lightning-fast/ Personally; I’ve always trusted in a high damping factor(around 1K, out to 1kHz, usually), SS amp and long Xmax, in a TL, for bottom. Lately; I’ve added DSP, to those.