"Szombaton nem itt vagyok" means I am not here (this particular place) on Saturday but somewhere else. It puts the emphasis on negating the place itself.

Slightly, yes. First it sounded weird. After some thinking, I can imagine someone saying it to mean "I am not available here on Saturday, only in my downtown office". A situation similar to "The bus does not stop here on Saturdays".

This sounded wrong first, but well, you could say it. This puts a strong emphasis on negating "showing up here in particular". I have a list of places where i will be but this one is not on it.

But still, I would rather use "szombaton nem itt vagyok"Depending on the place of "nem" you can negate a lot of things, similar to other languages, but