One of us thinks that "This weekend" and "next weekend" are the same thing (so as of the date of this post, June 21-22.)
One of us thinks "this weekend" is June 21-22 and "next weekend" is June 28-29.
Next
1 (Of a time or season) coming immediately after the time of writing or speaking.
2. Coming immediately after the present one.
What is the weekend coming immediately after the time of writing on June 16th? It's June 21-22.
On June 21, we will be in a weekend at that time. So what is the weekend coming immediately after the Present one? It would be June 28th and 29th.
So sometimes this weekend and next weekend are one in the same, and sometimes they aren't.
That said, the usage of this to refer to the upcoming weekend and next to refer to the one after that has crept into usage over the years, is widespread. Also, the closer one is to the weekend, the more likely people are to use this weekend to refer to the upcoming weekend and next weekend to refer to the one after that. Seeing as how it is always up to the speaker to convey exactly what they mean, when referring to next weekend, don't leave it to the listener or reader to guess whether you mean the upcoming weekend or the one after that. Make it clear. So again, sometimes this weekend and next weekend are one in the same and sometimes they aren't.
You'll see the same idea on a freeway. A billboard say 5 miles from the first upcoming exit will often advertise a place of business and say "next exit." Well it means that first exit. If that same billboard were say at an exit, it will often advertise and say Next exit meaning not this one that you're right up on already at the moment, but the next one. Or for example, you might come across a sign in Georgia that says Cartersville, next 4 exits, That means the first 4 exits you come to, not skip one and then the next 4.