What century was the year 2000 in?

[KoopaDaQuick 🇵🇸]

While 1999 is culturally known as the end of its century, and 2000 as the start of a "new millennium," in more formal and academic contexts I've always heard of 2000 as the last year of the 20th century and 2nd millennium due to the fact that there was no such thing as "the year 0" or whatever.

So, what is it? Is a century defined by the first two digits of a year, or is it defined by periods of exactly 100 years, including the 1st century?