It's a good question. There was no year 0, as far as I'm aware. We started counting at 1. So the first decade was 1 to 10 inclusive. That is to say, the year 10 was part of the 1st decade. Similarly, the first century was 1 to 100 inclusive, and the first millennium was 1 to 1000 inclusive. The second decade began in the year 11; the second century began in the year 101, and the second millennium began in the year 1001.
Does this mean that the year 2000 was part of the 90s? Well, it was part of the decade that began in 1991; it was also part of the 20th century; and it was also part of the 2nd millennium. However, when we say the 90s, do we mean literally the years with 199* in their year, or are we actually referring to the decade that lasted from 1991 to 2000? If the latter, then evidently the year 2000 was part of the 90s 'decade'.
Just as an aside, I agree with the Australian newspapers when they called their then-Prime Minister John Howard the "party pooper of the century" when he pointed out that the new millennium would not actually begin until the year 2001. Who really cares if it actually started in 2001? To go from years starting with 19 to suddenly years starting with 20 was an exciting event.