This is numeral pedantry, but it's all to do with cardinal and ordinal numbers. Anno Domini (AD) means " The year of the Lord", so the day Jesus was born, notionally, was the first day of the first year of the Lord. Hence 1 AD. There was no 0 AD because there is no such word as "zeroth" or "nothingth": these are ordinal numbers. However, BC dates are cardinal numbers, that is the number of years "Before Christ", hence there was a 0 BC.
Since the first year of the first millenium was 1 AD, hence the first year of the second millenium was 1001 AD and the first year of the third millenium was 2001 AD.
Just to confuse things, the latest estimates as to when Jesus was born indicate that it was the year known as 5 BC, as other records indicate that there was a census in Palestine in that year.
Myself, I follow the Chinese system...