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

From my time in the audio hobby and profession,  I feel speaker "speed" has much to do with the response time in moving with the very first note/electrical impulse.  The trailing edge of a note is easier for a transducer as it is already in motion.  However the leading edge begins with the speaker/transducer at "rest", and the possibility of the transducer being slower than the electrical impulse being sent to the transducer.  (sorry for the "transducer" word, but I do not put electrostatic and planer speakers in the same category as push-pull standard types of speakers).  

Interesting question and answers. 

When a speaker has "speed" it's approaching the sound of real, life like, sound

Designing speaker is a compromise - some are optimized for speed (dynamic), or soundstage, or musicality.  Some $80K speakers can have a huge soundstage but lack musicality.  Some are optimized for speed but soundstage may lack bloom.  And so on ...

I do not put electrostatic and planer speakers in the same category as push-pull standard types of speakers

That may be true, but conventional drivers are transducers nevertheless - they convert electrical energy to mechanical energy hence by definition they are transducers. And fundamentally there’s not much difference between electrostatic panels and conventional drivers.  They might be different but not fundamentally.

The issue of loudspeaker speed (like almost everything else about loudspeakers) seems to be endlessly complex.

Here's an interesting interview with Allan Hendry of Monopulse loudspeakers.

https://www.monopulse.co.uk/quest.htm

If what he's saying is correct, and I don't have enough knowledge to know any better, then many (if not all!) two way designs could be seriously flawed when it comes to speed, phase and integration issues.

Transient response, certainly, as has been pointed out by others, sonically perceived as a lack of smear/crispness, unforced clarity and liquidity. Usually, not least through quality all-horns, I find it linked to uninhibited dynamics as well (or: ease of "explosivity," felt as a startling experience), coupled to (at least as a partially responsible aspect) a pronounced degree of low volume level "ignition," whereby music comes alive quite easily at lower SPL's.