If you make it clear that that's the period you're talking about, I don't see a problem with it, but still admit the pedants have a point in that the name "the 21st century" *does* imply starting in 2001. You could use a different name (such as the convenient ones we had for the last few centuries, "nineteen-hundreds", "eighteen-hundreds", etc.), or you could just say what you mean and gain plenty of latitude (as in the Barbara Tuchman quotation at the top). If you try to assert that "the 21th century began on January 1, 2000" and that's simply what the term means, always, then I think you lose to the pedants on that :) Clearly there's basis for using that name for several different roughly-similar periods of time, one of which is "the years 2001 through 2100", and people do sometimes use it with that meaning and probably always will.

BTW, why shouldn't the first century include the year 1 BC? That would also be a "quirk of how we count years" :)