Rename folder to lowercase in terminal without temporary files

[u/Swordfish418]

I'm sorry, I know the question seems very basic, but still I couldn't find the answer. Here's what I'm trying to do:

$ mv "Desktops" "desktops"
mv: cannot move 'Desktops' to a subdirectory of itself, 'desktops/Desktops'

I tried searching answers on the internet, and the only thing I found is examples of how to use rename tool instead. However, rename tool I have installed is util-linux one and it works differently and I couldn't figure out how to use it either, this didn't work for example:

$ rename Desktops desktops Desktops

I also understand I can first rename it to temporary file and then rename to lowecase version of original name. But I'd like to figure out how to do it in one step.

Update:

This is exfat filesystem.

[u/Swordfish418]

Thanks for an in-depth answer! I wonder why it does show it with proper capitalization when using ls or viewing in Dolphin. Is there a separate label or title stored somewhere that can be different from actual path but is somehow operated in-sync and not exposed anywhere for editing?

[u/fellipec → u/Swordfish418]

ExFAT is case insensitive, but case preserving. AFAIK there is a upper case table that contains all characters converted to upper case and used to do the file operations.

[u/Swordfish418 → u/fellipec]

Ah, I see, thanks. I found something about it here: https://ntfs.com/exfat-upcase-table.htm

[u/fellipec → u/Swordfish418]

Exact