Profiles MOVED and BROKEN???

How did you configure the path to that external folder, separate from the default profile location?

When I created the new profile from the initial Firefox startup, or from about:profiles, I made sure to select "Choose folder" and then carefully navigated to the folder on the external drive. The problem happened when I wanted to "clean" the profile because it wasn't syncing properly (which I need mainly so that I can also get passwords etc on my Android phone). This is Macos Tahoe 26.5.2 BTW. From past experience, using "refresh Firefox" loses my groups (I use Simple Tab Groups extensively). Maybe I did something wrong but that has been my experience. So I thought maybe making a new, clean profile would work. I tried to backup the existing profile, make a new clean one, and then restore from the backup. But then not only did this not restore my groups, but also it had somehow changed the location of the existing profile to a default place under ~/Library/Application Support/Firefox/Profiles. However, then I could delete neither old nor new profiles because it said "Can't delete a profile that is linked to other profiles". In the end, I manually edited ~/Library/Application Support/Firefox/profiles.ini to retain only the old, possibly problematic profile. I'm not sure that I recommend this but I am quite used to this sort of thing and if the worst comes to the worst of course I have backups. Also, as others have suggested, this is quite a simple way to re-use an existing profile for which the files exist (eg., from another computer, on an external disk) but Firefox doesn't know about it. Again maybe not the recommended way but it has worked for me. Now, not only is the old profile syncing perfectly but I have it as a want it. But there is now an odd addition to the icon on the Macos dock. It looks like a tiny closed book symbol. I don't know whether there is anything to worry about.

PS: I said "sync is working perfectly" but although it seems to work, it isn't saving sync logs, as seen with about:sync-logs.

Okay, that way should generally be safe. But you're right once you start "managing" the profiles, it will start using the default location again for any troubleshooting steps like the "Reset" feature, so that's expected. Basically the answer to that is, create a new clean profile in the external location, have a backup of your existing profile folders, remove the other profiles via about:profiles, so that you only have one, clean, in the external location, and then copy over the contents of the profile folder from the backup one to the clean one — while this undoes any "Refresh" functionality, as you're basically putting back your profile 1:1 with no cleanup, this way you can at least prune the profile list.

As for the icon badge — that means there's more profiles, once you remove any surplus ones you don't need, and you're down to one profile only, the badge will be gone.