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

Here is a question: if both speakers have the same frequency response, why does one speaker have better speed vs the other?  

Another question: if one speaker uses a 4in. driver for the midrange, the other uses a 6.5in driver for the midrange, and both have the same frequency response, which will have more "speed"?

It's difficult to say since the 4in driver might have better speed in the midrange, but the 6.5in driver might have better speed for the bass.  

Also in general, if a speaker has a tipped up treble response, it does lend to an illusion of having better "speed" especially on the leading edge such as guitar.  Of course having too much leading edge could be a problem too.  

Yes.  The amplifiers output impedance plays a role.   It along with the speakers impedance determines the Damping  Factor which determines how well the amp can control the movements of the speaker's transducers/drivers.

https://www.sweetwater.com/insync/damping-factor/

I think our perception of speed has more to do with how fast a speaker driver stops making sound after the signal stops than how fast it reacts to a signal. That's why larger drivers often sound slower than smaller drivers. It's not too difficult to get a driver to respond quickly when its voice coil is energized. It's a lot harder to get it to stop vibrating once the signal goes away. 

Servo controlled drivers can overcome some of these limitations because they can use power to stop the driver quickly when the input signal stops. 

Another way of talking about "speed" is "lack of stored energy."

That is, the waterfall plot is very clean and very short, without ringing.

Not to brag, but...

My subwoofers are so fast, they can do a 20-cycle sine wave in a MERE fifty milliseconds.

Not only that, I’ll go out on a limb and claim NOBODY makes a subwoofer that can complete a 20-cycle sine wave in less time than mine.

;^)

Seriously, I agree with those who have noted in various ways that what happens with the trailing edge of the note - how quickly and cleanly it decays - is (in general) more important than what happens at the leading edge of the note, at least when it comes to the subjective impression of "speed".

Duke