So, you celebrated the beginning of the third millennium, don't you? But let me tell you, you are wrong.
There was no "Year 0". The first year was called "1", though they did not do so in the ancient times. But several years later, all historic data was put in a line, and it was decided to start the new counting with 1.
The second year started when the first had finished. That's easy to understand, isn't it?
The second decade (10 years) started, when the first decade had finished. Let's do some counting: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. So the second decade starts with year "11", which is easy to understand, isn't it?
Similar, the second century (100 years) started with the year "101". Now, which is the first year in the second millennium (1000 years)?
Yes, it is 1001. As you calculated correctly, the third millennium starts one thousand years later, in 2001.
So why does all the world celebrate the new millennium one year too early? I don't know. Must be a mass psychosis or something like that.