Millennium 2000

You clicked the button to say that the millennium starts on 1st January, 2000...

Congratulations!!

You're absolutely correct!

If you count backwards with our modern calendar, and regard the first millennium as starting on 1st January year zero, then 1st January 2000 is exactly 2000 years later, and obviously the start of the "third millennium."
There are some minor problems with this: the most obvious being that there is no particular significance to the day which corresponds to 1st January, year 0. No-one at the time could possibly been aware of the fact that it was "year 0", since the count wasn't assigned until a few hundred years later, notionally from the birth of Jesus; nor could anyone have known that it would have been 1st January if a calendar finalised 1500 years or so later had already been in effect. It seems clear historically that actually Jesus had been born three or four years earlier.
The most commonly cited "error" in this view is that the year 0 "didn't exist". Well, the year before the year 1 certainly existed: it's just that some rather innumerate person labelled it "1 BC". The astronomers, at least, have apparently got tired of this arithmetic slip, and do indeed refer to this year as exactly 0. The previous year is thus -1, and so on. Rome was (notionally!) founded in -752, or "753 BC" in the other notation.
One significant date that this view does match up to is 1st January 1900, when it seems there were celebrations for the start of the "20th century": so unquestionably 1st January 2000 is the centenary of this date. The other useful feature of this view is that decades and centuries fit into place rather easily, with convenient names like "the nineteen-twenties", though oddly no-one seems to know what to call the next couple of decades.

Most sensible web reference I've found yet

When Does the New Millennium Begin? by Peter Meyer
And the bottom line? If someone you suspect of pressing the other button invites you to a party - just go!!