As an example of what I do, when I generated my current fortress and adventurer gameworlds (same one, copied prior to either embark, so as to be used concurrently/independently for each game mode) I made one "region1.fort" and the other "region1.adv".
It's still "region1" for both. And this more or less comes from when I've had a worldgen I've felt happy copying into a newer version of DF I might have ensured that the "regionN" from the original game is now "regionN.legacy" to avoid inadvertent clashes (renaming any imagedumps/etc likewise). And I've sometimes saved (one level deeper, in a "_region1" folder within which I can save maps[1]) a copy of "region1" as "region1.raw", which I've been known to re-copy back up to replace the "region1" that I've totally messed up on embarking.
The "_region1" folder, itself, arose from being a handy place to hide the copious Seasonal Backup files that get created. Whether or not I ever plan on rewinding time, I like the security of having them around but after a few years they clutter up the loading menu. And I sometimes think I have a pathological fear of deleting data, so a quick drag of anything older than 2-4 seasons prior to the last seasonal is therapeutic without causing injuries to my psyche. The underscore is just what I use to make it obviously different. Calling the folder "region1.backups" or similar would be as obvious, but just not my way of doing it...
I could just as easily change the "region1" bit, as well, to something meaningful. Unless I've deliberately gone for a particular fortress name (or taken the random one at face value, for some reason) I'm probably not likely to consider the "English" translation meaningful at all, but horses for courses I suppose... But (for OP) having "waterfall" or "volcano" or "ZOMBIES" as a new name for any given "regionN" (with or without further prefixes/suffixes, as required) wouldn't be too much of an effort.
Although I tend to do renames, if at all, while DF is not loaded or is at least back at the main menu screen. As a caveat for OP's suggestion might I suggest that there be a "save manager" option at the loading screen (at least!) that lets you do the renaming of "region1" without needing to use the Windows/whatever file manager?
Maybe it would be a sop towards excessive save-scumming if it allowed (with appropriately coded safety interlocks to ensure a 'clean' copy, and possibly keeping synchrony with the "<foo>-Aut-1053"-like autosave folders?) one to rename/copy/copy-into-nonstandard-locales a game while even running, of course, but definitely I would make minor, but significant, use of a load/start-game menu save-folder manager.
[1] Like the ones I edit to keep track of quests both informed about and complete; and the little text file full of useful notes about the world, etc...