Well Wolf, "speed" is another of those audio terms that is hard to define other than to say you certainly recognize it when you hear it. Much of it to my ears is MOST readily recognized in large scale acoustic and choral music with wide and varying dynamic shadings and tones. A "fast" amp seemingly resolves the leading edges of instruments without softening or rounding off, an effect I hear primarily with OTL tube amps, a greater transparency and resolution of all the musical information on the recording. The ability to resolve dynamic shadings without the sound congealing or dynamics compressing. Now as noted by Charles1dad there are amps that are considered "fast" in this manner but miss the subleties of tone and true and accurate pitch of instruments which can, to my ears, lead to fatigue. When the pitch of instruments sounds natural and more like what you hear live and the subtle dynamic shadings and sense of space is preserved, it just sounds more real.
A good test is listening to large scale music at lower volumes and hearing all the information, including bass, preserved as it is at higher volumes. This is definitely an area where "fast" OTL tube amps excel over most if not all transformer coupled tube amps and SS.