Not A Poet
Not that it really makes a great deal of difference in the long run because, after all, all we're really talking about here is just our calendar, not absolute time at all.
But to go back to your argument, if you were born on Jan 1 of the very first year of our current calendar then the absolute second you became one year old would have been on Jan 1 of year 2 and the second you became 10 years old would have been on Jan 1 of year 11. The calendar started with year 1 rather than year zero. You can count it out on your fingers (figuratively of course) and see that the second you became 2000 years old was on Jan 1, 2001. So that is when you actually began your third millenium.
Since you were born on the very first day of the calendar, you are exactly as old as the calendar and thus it also begins its third millenium on Jan 1, 2001. Jan 1, 2000 was also a significant day in that the calendar rolled over all four digits from 1999 to 2000. Those of us who need to get the maximum good from our cars have experienced the "thrill" of watching the odometer roll over from 99,999 to 1000,000 miles. That's always more interesting the the roll over to 100,001. Much the same concept.
Well, that's my $0.02 worth.
Imagination is more important than knowledge