I guess the above description is exactly what "touch sensitivity" is all about. I've always sought the "correct meaning" of exactly what "touch sensitivity" is...and I think Sekunda nailed the definition with his description of how the amp's tone interacted with his playing attack, as described above. Thanks.
Very good guess, Bennett! I think it's actually a very good description, too. I'd like to give a little background on it just for grins, because it's possible that the term means different things to different players.
Strictly speaking, in audio-land the term "gain" simply means signal amplitude. The fingers and pick act on the strings to vary the amplitude acting on the pickups and coming out of the pickups from the guitar's electronics.
Thus there are two distinct elements to this touch-sensitivity thing: one is the ability of the guitar to reproduces nuances the player can create with pick attack, fingers, etc; the second is the ability of the amp to respond to those nuances and changes in amplitude, and reproduce them in a musical way. Sounds easy, but isn't when it comes to making things sound really good.
If the guitar's volume and tone controls are maxed, the guitar's own headroom can be compromised, depending on the instrument. Whether or not this is useful will depend on the player's style.
Any amp will have a little less touch-sensitivity if the guitar is essentially putting out one volume level, or if the player is using a compressor pedal set pretty high.
But if the guitar can produce nuanced tones, a touch-sensitive amp will respond by musically responding to the changes in signal, gradually allowing saturation in its internal tube pathways to allow the signal to clip when pushed hard, and simply saturate in different ways when pushed lighter to varying degrees.
This is really what touch sensitivity is about, IMHO. All tube amps are touch sensitive to one degree or another, the question is how well they do it. In other words, gain shouldn't be "on" or "off" the tubes should saturate in a variety of ways depending on amplitude.