Being too naive and reading only the first 4 entries, and believing in how easy it was, this update became rather frustrating, just like when installing Big Sur at the start. First, there was no 11.1 update found. Had to go to the App store, look for Big Sur, then press the GET button. Then it took me back to some similar System update menu, but it was different looking from the original one. If it wasn't the official App Store, I would think it was some Virus installation. It took about 1/2 the total time before rebooting.
At the restart menu, I noticed "Macintosh HD" pre-selected, trying to be too smart I changed it to another drive with a "better" sounding name, and it became a restoration of 11.0.1 instead.
After learning my lesson, I went back to read more entries, thanks to BustyCat's entry, I repeated the entire process, and this time leaving everything alone (keeping on "Macintosh HD"). It took a few reboots, and the installation went smoothly afterwards. I used 0.6.3.