"This Saturday" surely does mean the closest Saturday in the future. But "next Saturday" has become ambiguous. I've been asked more than once when I used the expression whether it's meant to be the Saturday in this week, or in the week after. (I would usually mean the week after, but if today is Sunday, I might use "next Saturday" to mean the Saturday six days from now. So I find myself being inconsistent.)
Even more subject to confusion is "the Saturday after next", which I'd assume means "the second Saturday from today" -- but I'd want additional confirmation, since not everyone seems to make the same assumption.
Regarding "this Monday" and "next Monday" (or indeed any day of the week), the interpretation does seem to differ depending on what day it is today. If you're making an appointment, I'd suggest confirming the date.
Edit: Inspired by comments, the closest next Saturday can also be identified as "this coming Saturday", and the next following Saturday, as "Saturday week" or (as I learned it) "Saturday a week". As the comments indicate, accurate identification still depends on a (possibly explicit) agreement between speaker and listener.
(Bias: native speaker, of an "older generation".)