In North America, calendars usually begin the week on Sunday. The calendar I bought in Spain starts the week on Monday (slightly disorienting for me). Computers often have an option to choose which one you prefer.
To the OP, 6 days from now. “Saturday next week” would also have meant 6 days from now, for me the week begins on Monday.
Huh - this gets worse when you are in a country that does not have Eglish as a first language - life is fun.
Slight hijack - when you refer to stuff that you “did” or will do in the weekend should you say
a) at the weekend?
b) in the weekend?
c) during the weekend?
e) over the weekend?
a) - bugs me, wekkend is not a place, how can i ever be “at” it?
b) - logically should be the same complaint - but the useage is comfortable for me
c) - my preferred but always feels slightly pompous
d) - we also used to use a lot, but logically should not be correct
BTW - here in Singapore most people use the “at” construction
Another slight hijack - since when did “troops” come to mean a single soldier? As in “three troops were charged with Abu Gharib abuses”
I always thought that troop was a collective noun
i.e a troop of soldiers (a group of soldiers)
a boy scout troop (a formal grouping of boy scouts)
I wish those Egles would just go back where they came from 
In North America we say on or over the weekend; in Britain they say at the weekend.
And, fortnightly, which I use to describe things which happen every other week.
Six days. The next occurring Saturday.
Fromage A Trois, matt_mcl, Antigen, Antinor01, elfkin477, Nava, Clothahump, you are wise and sensible people. 
I take it as the next occuring Saturday.
This can be modified if you first use ‘this’ as in:
This Saturday we will go to the beach, then next Saturday we will visit grandma.
But this can be further modified a 3rd time:
This Saturday we will go to the beach, then the next Saturday we will visit grandma, then the next Saturday we will go shopping.
So in my usage next means the next occuring after prestated conditions. and can actually go on some time in the future:
The 1st Saturday in January we will leave for our ski trip, the next saturday we will be heading back home.
I also use the term ‘the following Saturday’ to state the week after next.