Thank you for your detailed response. Yesterday I spent 10 hours to be able to fix this issue.. During that time, I even learned the basic commands of Terminal!
Without seeing your comments I also discovered the same procedure would help me. So I first learned how to open the recovery mode, then there I found out how to use Terminal. Then I checked for codes to see the disk space etc.(a lot of research...) As in your case, my Macintosh hard drive was also full. So I connected an empty 500 gb external hard drive. By using coding I tried to copy files but it didn't work. By accident, I discovered Terminal> Shell > Open Window. Then I could see all my files/ folders in my Macintosh HD, however I couldn't move them. Then I decided to install Mac Big Sur on my 500 gb external HD. But first I had to learn how to format it the proper way (omg so much thing I had to learn in one day!!) Through one page, I found that it has to be APFS (Apple File System) but there was no such thing in the options.. Then through another page, I learned that I have to choose GUID partition from the drop down menu. After doing that, I could choose APFS.
Anyways, after erasing my external hardrive (so formatting it as APFS), I also installed Mac on that External Harddrive and could open up my laptop again. However, I didn't see my old files (as it became basically a new computer on external HD). So, to be able to see my files on Macintosh HD, I again opened the Terminal and then looked up for codes to be able to see files. After finding the files, I could free up a lot of spaces (using delete code for the folders I want to remove). Then, I closed the computer, remove the external hard drive, and tried to open it (by thinking that it will now complete the install as there is now a lot of free space). However, it didn't complete. It was in a loop so many times. That's why I again opened recovery mood and this time chose Internet Recovery (one youtube video was describing that this option keeps your files, however there might be a risk.). Anyway, when I chose internet recovery, the computer asked me where to install Big Sur (there was only one option (Macintosh HD) as I removed external HD and it showed that now I have 75 GB free space which is more than enough. After an hour or so, It installed Big Sur on my computer. However, I lost all my data :((( luckily I don't remember having very important files. Eventually, this was not a good experience for me even though it is not the end of the world. I really wanted to rest this weekend and I spend my whole Saturday with this **** :(
I don't know what that "looking glass" option you are referring to. Maybe if I knew it, I could also copy my files (because I could see them through the terminal, but couldn't copy them by coding. They were not visible in the Finder.)
Anyways, from this experience, I can totally say that Macbook and update procedures are not good. My completely fine working computer got into this mess for nothing.. They have to inform users that at least 35 GB space is needed, not 12 GB!! I lost all my data.. Such thing I never had in my Windows computers (the current one is 8 years old, working perfectly..) Please Apple pull yourself together!!!