The question is interesting indeed. If I understand correctly, there should be also lines like "installing component 'rustc'" (and so on for every downloaded component), not only for 'rust-src' and 'cargo'.
Maybe.., but seems not. Always the same picture. I have saved screenshots (1.60, 1.61)
Next time I will use 'rustup update stable >> up.txt' and post here
Directing the output to file is a good idea if you wan to save the text. But keep in mind that the ">>" will append to the up.txt file if it already exists.
Also the output of rustup is to both stdout and stderr.
If you do the command as you posted it will only save the stdout stuff and the stderr stuff will still go to your terminal.