Your forefathers would not have agreed. I have in my possession two or three newspapers dated 31 December 1900, and they all refer to the issue as being the last issue of the nineteenth century. So the twentieth century began on 1 January 1901 and consequently the 21st began on 1 January 2001.
There was no year 0. The year which we refer to as 1BC was followed immediately by 1AD (in historical references). Astronomers use a different numbering system, in order to make calculations easier.
Labels such as the twenties, sixties, etc. are usually used in a style context and often refer to part of the decade only and might even overlap into the next decade (or from the previous one); for instance, what is popularly meant by the sixties usually only refers to the period (about) 1963-70, since stylistically and culturally the early part of the 1960s is indistiguishable from the late 1950s.