I’m not sure I entirely agree with that description on the Blackstar site.
Gain definitely is about driving the input to the pre-amp stages and getting pre-amp overdrive. But the channel volume is the volume of the output of the pre-amp, and shouldn’t have a significant impact on the tone (with caveats).
The clue to the function of the channel volume is in the text: “If you have a single channel amp like the Studio 10, you will just have a single volume”
This indicates that channel volume is something that’s specifically a function related to having more than one channel on the amp.
To set some context, traditional guitar amps did not have “patches”, but might have multiple channels. These channels, typically, have different pre-amps gain structures (so you might have a clean channel and a “dirty” one).
In some case, these channels would have completely separate pre-amps with separate physical inputs, so to switch from the clean channel to the dirty one, you would have to unplug your guitar and plug it into the other input.
The idea of channel volume in this environment is to balance the output levels from these different channel pre-amps into the shared power amp. This is so that when you switch channels you don’t get a big change in volume.
With many modern amps, like digital modelling amps, you don’t have separate channels but, instead, have “patches”. However, these perform a similar function to channels, but they are software-based rather than being hardware-based.
Cheers,
Keith
Addendum: Some traditional amp circuitry may have characteristics such that the channel output level does impact the power-amp gain differently from the how the master volume impacts it. This might be due to, for instance, where the tone control circuit is in the amp circuit.
In those cases, the channel volume may have an impact on the tone, but this tends to be specific to certain amps.