Rather than TimeMachine or iCloud, which files do I need to backup on a external hard drive?

I think a dedicated external drive for backups is necessary. Apple recommends not sharing other storage on backup drives, and some have reported losing those other files when doing so (especially under Big Sur -- I don't know for sure if this has happened with Catalina).

I will share a couple of thoughts. First, to be clear, iCloud is not for backing up and it should not be relied on as such .While it does have some back up features, it is for cross platform sharing.The rest of your comment is a bit confusing. Why do you not want to reformat your Time Machine drive? Yes, you should backup prior to installing the OS upgrade. When backing up, you want all of your files to be backed up. Why would you want to have some at a risk of being lost? Do I assume correctly that your Thunderbolt drive is used only for backup? I would recommend that. Time Machine normally is fully capable of doing full backups as you would expect. Beyond that, I am not sure what your question might be. I hope this is helpful to you.

As Ronasara points out, a "good backup" would be one that backs up everything and enables you to recover your entire drive should something go wrong. Examples of such backups include Time Machine, which provides a versioned backup enabling you to recover earlier versions of your files and system from earlier times, and "clones" which can be bootable that provide one single mirror image of your entire computer at a given moment intimate.

Most of the files that may matter to you are probably in your Documents folder. Some may be in the Desktop folder and Downloads. Items in Pictures (and Movies) are probably important too. Music is in a different folder. Your Apple Mail can be recreated but to do so requires exact copies of files and folders in various different folders on the Mac, some of which are not easily visible to all users. Saved logins and application serial/registration numbers are saved in other places, depending on the application. iPhone backups are stored somewhere else, if you have done those on your Mac. To manually try to copy and save all these things successfully is very difficult which is why most people simply use the complete backup solutions offered by Time Machine and various "clone" utilities (SuperDuper, Carbon Copy Cloner are two examples). Time Machine is free, the other utilities may have free versions available with limited functionality. For backups, I get the full function versions and keep at least two backups with different methods, e.g. Time Machine plus an independent method. The reason for having redundant and independent backups is seen in these Discussions, where sometimes people have trouble with their Time Machine disks or with the disk that has their "clone" backups. All disks do fail eventually, none last forever. In fact, a power surge that happens during a backup could cause both the computer and the backup drive to fail at the same time. Rare, but it does happen.

You absolutely should have complete backups (more than one) at all times, and especially before modifying your operating system.