This is going to be related to
xdg basedir specification. Something, probably Waterfox itself, is interpreting these variables (see also
XDG user directories on the Arch wiki) and generating XDG_DESKTOP_DIR which by default (i.e., if left undefined) is ~/Desktop.
For what reason it always generates this is a very deep topic, but it probably includes mundane things like, "Check if we already have a shortcut on the desktop" in the same vein as "Are we the default browser," or perhaps "Need a default location for downloads" and your always-save-to location in Waterfox settings might be this Desktop directory.
I don't have any creative ideas about which specific variable or which specific value will suppress this behavior, because the exact reason why it wants to make that directory is unclear. But if you play around with these and keep starting Waterfox Classic over and over again, maybe you'll hit a winning combination.