This is going to sound extremely silly, but I am under the possibly false impression that when I bleed my brakes, I am also bleeding the fluid that operates the hydraulic clutch. Is this correct, or false?

I'm honestly not sure.

*hides behind a door, avoiding imminent tomato throwing*

 

we have something like a double reservoir. but the clutch fluid should remain fine even tho you are doing the brakes. yes its the same fluid and all but the brake fluid will run out before the clutch will because its drained from a higher position or something. just investigate the reservoir and you'll figure out what i mean

 

same reservoir, different lines/system they are bled independently, and the procedure is quite different...DO NOT PUMP THE CLUTCH PEDAL FOR PRESSURE!!