An error occurred loading the update - Big Sur installation

I would probably use either rsync -a or cp -a, but you will need to mount the internal disk first.

To be honest, I'm not great at Terminal without doing it myself and often Googling a bit on the side, so while looking for an appropriate guide, I came across another suggestion, which is to make a disk image. This has the advantage of baking up everything, so you can be sure to not have lost anything. The only disadvantage is that you will need a drive big enough. You can do this with the Disk Utility, so no need to use Terminal.

However, if you want to just back up the home (/User) folder, the same thread gives details on it. The only thing I would do differently is use archive mode (cp -a) instead of just recursive, as this will retain more information about the file structure. More information about cp.

The thread with the instructions: https://apple.stackexchange.com/questions/274032/backup-to-external-in-recovery-mode-terminal-or-time-machine

Invest in an external hard drive and, using the Terminal in recovery mode, back up the /Users directory and anything you think you need from /Library (probably nothing, most things are in ~/Library). Then wipe the disk and reinstall, probably from USB.

If you can’t risk losing the data on the disk, you should have a back up anyway. At least this is happening at a time that you can still retrieve the data, as opposed to something where you can’t, like if it got stolen.

On mine it's called Data, not Media, but I think it's the same thing. That's the one where your files are stored though, so make sure you back that up.

For me, the update didn't work until I arrived the entire drive, not just the volume. To do this in Disk Utility, you have to go to View > Show All Devices, and then wipe the entire disk before reinstalling the operating system. Obviously this leaves you will no files whatsoever, so you absolutely need a back up that you have checked works and contains everything important.

I keep almost everything in either iCloud or Dropbox anyway, so I can never lose anything too precious.