You clicked the button to say that the millennium starts on 1st January, 2001...
If you count backwards with our modern calendar, and regard the first millennium as starting on 1st January year 1, then 1st January 2001 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 1. No-one at the time could possibly been aware of the fact that it was "year 1", 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 four or five years earlier.
The only other problem with this view is the labelling of decades. Either we have to divide centuries up in a rather odd way - the 20th century consisting of a nonade, from 1901 to 1909, followed by nine decades, the 1910s, 1920s, and so on, and a totally odd year 2000, or be consistent. Well, good news: the United Nations Committee for the Rectification of Astronomical Peculiarities (better known by its acronym) is now looking at a radical proposal to aid consistent decade labelling. This year, formerly known as "2000", really belongs to the 1900s, so will be renamed "nineteen ninety-ten". So that computers and things can keep up with all this, the hexadecimal symbol 'A' for ten will be used to indicate this final digit. (Those with programming experience will spot that dates will thus still sort in the correct order in the new representation.) Here, then, is the calendar for the four decades from 1991...
1: "Nineteen ninety-ten" - ends the "nineteen-nineties", obviously.
2: "Twenty-oh-ten" - the decade is the "twenty-ohs".
3: "Twenty-tenteen" - the decade is the "twenty-tens".
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!!