What century was the year 2000 in?

[Mr. Morden]

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?

Both.  It’s any period of 100 years.  So, for example, 1900-1999 is a century, commonly referred to as the 1900s, while 1901-2000 is a century, commonly referred to as the 20th century.