In the comic above, Ponytail accuses White Hat of being pedantic. Hah. I'll show you pedantic.
The 21st century began on the first day of 2001. This has nothing whatsoever to do with whether there was a year zero. It's just how counting works. If you're eating donuts like mad, the first dozen comprises donuts number 1 through 12, and you don't get to the second dozen until you start on donut 13.
(If you think nobody ever eats a dozen donuts, you've never been to Britt's Donuts in Carolina Beach, North Carolina. On one such visit I watched a girlfriend consume over a dozen when she initially insisted she only wanted one. More generally, you don't have to eat a dozen donuts all at once to eat a dozen donuts. You could eat them over the course of month or a year or whatever.)
Also, by the straightforward meaning of the words, the 2000s began on the first day of the year 2000, and that's true whether you're talking about the 2000s in the sense of a period of a 1000 years, 100 years, or 10 years.
Likewise, the 2020s began today, the first day of 2020, but the third decade of the 21st century won't begin until the first day of 2021. That would also be the start of the 203rd decade, if the 203rd decade were actually a thing.
It really is that simple. What screws it up is the assumption that the 100-year period we call "the 2000s" and the 21st century are synonyms. They're approximately the same, they're the same to within one percent, but they're not exactly the same.
Also, a century can be an ordinal century (as in "the 21st century"), but the term also means any 100-year period. So 2001-2100 is a century, and so are 1951-2050 and 2000-2999.