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

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.

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. :)

Agreed, that makes sense to me.
(And resonances in a speaker can of course change perception of tone).
I have a pair of old Thiel 02 speakers which were cheap even circa early 80’s. It’s a good design making the best of a cheap box and cheap set of drivers at the time.
But boy does it sound "fast" in the usual audiophile sense of having transients pop out vividly, and having no bass overhang. They are rhythm-machines in that regard.
It’s just that they have a certain frequency balance, very flat in the bass region and maybe a tiny dip in the right places, to have a frequency response that creates this impression.

At least, from what I can infer.

Speed refers to transient response. 

The ability to follow musical transients (percussion, strings plucking, etc) closely. Both the attack and decay are important. 

Speakers with lighter, stiffer cone materials, and strong magnet structures are usually  have good speed.

Ribbon, electrostatic, and planar magnetic drivers tend to have good speed.