Thank you for a tip, but... at least at the moment I was not able to get it properly installed...
Perhaps the problem was, that in my PC other versions of Proton were not in that directory that your guide mentions, instead I found Proton versions in folder:
~/.var/app/com.valvesoftware.Steam/.local/share/Steam/steamapps/common
Then I was not sure how to "install" it manually (simply copying extracted folder to steamapps/common had the same final result as using "protonup"), so I found some commands from the internet, I did following commands:
sudo apt install python3-pip
pip3 install protonup
protonup -d "~/.var/app/com.valvesoftware.Steam/.local/share/Steam/steamapps/common"
As a result, I do have folder:
~/.var/app/com.valvesoftware.Steam/.local/share/Steam/steamapps/common/Proton-6.21-GE-1
But even after several restarts of Steam, it does not seem to recognize its existence. I thought I was "clever" and switched between Proton versions and restarting Steam... But even after doing that about 5 times or so... Proton-6.21-GE-1 is still not in the selection list...
I will probably try to do something again tommorrow...