Ooooh, interesting ...
I think we need to distinguish between two different uses for "desktop" here - desktop PC and desktop case.
Nitpicking? I don't think so. The term desktop PC is a catch-all for a non-mobile (i.e. not a laptop) PC. I remember back when my parents got their first PC pretty much
all PCs came in a desktop case - it would've been really unusual to have a tower. The term obviously had enough traction that it continued being used for the entire category of non-mobile PCs even when tower cases started to become more common.
That makes use of the term "desktop" on its own ambiguous - you could be taking about a desktop PC, which may be in a tower case or a desktop case, or you cold be talking about a desktop case. Whereas the term "tower" would pretty unambiguously mean a desktop PC in a case that was taller than it was wide.
Of course, you then get into the various terms people use for a base unit (that, a desktop PC without any peripherals or monitor) - which may include things like "box" and, surprisingly, "CPU".
Basically, all computer terminology will tend to some degree of ambiguity. Probably time to cherish our differences and move on