Good question.  In the United States this is a frequent source of confusion.  I always try to clarify the date when someone uses the term "next Saturday" so that I can be certain that we are referring to the same day.

My usage:

The Saturday that occurs in two days:  this Saturday or Saturday

The Saturday that occurs in the subsequent week:  next Saturday

Thank you to all the members who contributed to this discussion...
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Su.Ki.

It appears that there some common consistencies and a few differences.

I posted this discussion on Thursday, July 7 and I was asking about Saturday July 9.
Here is how I would refer to Saturday July 9...

1... I'll see you Saturday

2... I'll see you on Saturday

3... I'll see you this Saturday

4... I'll see you this coming Saturday

5... I'll see you in two days

6... I'll see you the day after tomorrow.

The confusion, if there is any, would arise when we say next Saturday or Saturday next.
If someone said next Saturday or Saturday next, I would not be sure if they meant in two days time or in 9 days time.

The dictionary defines next as: immediately following in time.
So that would suggest that Saturday the 9th would be the 'next' Saturday, yet I would not feel comfortable saying next, especially in this case, because of the proximity of the 7th to the 9th.

Ohhhh, the English language...

today is Thursday so...Scotland

This/On Saturday (in two days from now)

Next Saturday ( a week from this Saturday)

Next again Saturday (not this Saturday or next Saturday but the one after)...Essentially two weeks from this Saturday but this is how we say it :-), Could be different in other parts of UK

On a Thursday, I wouldn't refer to 'the day after tomorrow' as <em>next Saturday. </em>I'd call it <em>this Saturday </em>or even just <em>Saturday. </em>For example, if, during a conversation on a Thursday, someone said 'I'll call you on Saturday', it would be clear that they meant in two days' time.

For me, <em>next Saturday </em>would be the Saturday nine days' hence.

For example, if we had this conversation on a Thursday -

 'John's managed to book a Saturday flight.'

'So when's he going? Next Saturday?'

'No, this Saturday.'

- wouldn't it be clear what we meant?