The last strikes me as turning on an equivocation. A century is 100 years, and any 100 years can be a century; for example, 1815-1915 makes a plausible historical unit of 100 years, more so than 1801-1901. But "the twentieth century" means "the twentieth hundred-year interval in a series of hundred-year intervals with neither gaps nor overlaps that make up the Common Era." And the first day of the Common Era was January 1, 1 C.E. The day before that was December 31, 1 B.C.E.

It's a convention, but it's a convention that makes numerical sense; it doesn't require introducing a fictitious Year Zero, or having a ninety-nine year "century," or skipping over some number of years somewhere along the way. So it's a convention I find reasonable.