The solution to this problem clearly requires the assistance of good ol' Merriam-Webster.
THIS: 1 a (1) : the person, thing, or idea that is present or near in place, time, or thought or that has just been mentioned <these are my hands>
NEXT: 1 : immediately adjacent (as in place, rank, or time)
Clearly, "this" Saturday means "Saturday which is near in time." Similarly, "next" means "Saturday which is immediately adjacent in time". Now, unless time has become some crazy non-linear thing which jumps around from point to point, the Saturday "immediately adjacent in time" to Sunday would be in 6 days.
Using "next Saturday" to refer to the Saturday after next is a colloquialism, plain and simple. Colloquialisms are what happens when people say things because they "sound right", rather than thinking about whether they make grammatical sense.
Yeah. I'm strangely passionate about grammar. It confuses me, sometimes. [The passion, that is, not the grammar. Whoops, sentence fragment!]