[randcraw]

Is it? The first century spanned year 1 to 100 (there was no year zero). So the XX00 year is considered to be part of the preceding century. Thus 2021 is indeed the 21st year of the 21st century.

[bananapear]

If 2000 is part of the previous century, was 2001 actually the true start of the new millennium?

[randcraw]

That's the standard practice, I believe. Presumably that's why Arthur C Clarke chose the year 2001 for the name of his book rather than 2000.