The term "gain" as applied to electronic circuits has well-defined meanings: it is the amount by which a circuit increases the input voltage or current, expressed as a ratio or in logarithmic (decibel, aka "dB") notation. In most audio applications - including electric guitar - the term "gain" when applied to a complete amplification system refers to voltage gain: the ratio of output voltage to input voltage. You can't drive a speaker with the output of your guitar; its output must first be amplified. As an aside, gain is always frequency-dependent, which is another way of saying that a device's magnitude response is not flat. Gain may be constant over a wide range of frequencies - e.g., in a quality power amplifier - but it is never the same for all frequencies.
Practical amplifier circuits always consist of multiple active stages, each one of which produces some amount of gain. The apportionment of gain among the various stages is known as gain structure. Gain structure determines the noise performance of a system as well as the relative clip points of the stages. If you want the output stage to be able to deliver the maximum possible power without clipping to the load (speaker in our case), you need to have enough gain margin in prior stages to make sure the output stage is the first one to clip. Almost all, if not all, of the original guitar amps were gain-staged in this way: as you turn up the volume control - which actually reduces attenuation of the signal at that point in the chain - the first stage to clip is the output stage. Believe it or not, the objective of all these designs when they originated was to minimize distortion and noise.
As guitar players increased their playing volume - in many cases because they were playing in ever-larger venues - the distortion that inevitably appeared was more and more noticeable and gradually became an accepted part of the music they played. No guitar player ever gave much thought to gain structure, nor would it have mattered, as there was no easy way to change it. Master volumes didn't yet exist. There were some tricks played to get more distortion in recorded sounds without endangering the amp or speaker. The appearance of fuzztone pedals came about in the mid 1960s, and clean boost devices first showed up ca. 1971 (e.g., the Electro-Harmonix LPB-1).
Enter a bit of historical (non-)trivia: the band "Mountain" had a deal with Sunn and used their amplifiers. In advance of one tour - they were constantly touring ca. 1970-71 - they were supplied several Sunn PA heads, which were tube-type devices. The heads had several inputs, and each input had its own volume control. To control the overall level of the mixed signals, there was a - wait for it - master volume. Leslie West tried playing guitar through one of these heads and quickly discovered that he could get more overdive/distortion at lower volumes by running the channel volume high enough to clip downstream stages while adjusting his volume with the master. And the rest, as "they" say, is history....
I wanna know what gain is / I want you to show me
