Yes, there are a hundred years in a century, but there's no reason a century can't start with a year ending in 00 and end in one ending in 99. Since there was no "Year 1" any more than there was a "Year 0," there's no real basis for the claim that a century has to begin with '01. It's just an arbitrary belief based in some incorrect assumptions about the origins and history of the calendar.
And it actually makes more sense to start a century at a '00 year instead of an '01 year. It lines up with how we speak of decades. We talk about the Twenties, the Thirties, the Forties, etc. So we count a decade like the Twenties as being 20 to 29, not 21 to 30. Once '30 is reached, it's not the Twenties anymore, it's the Thirties. So if it's customary to begin a decade on a year ending in 0, then logically a century, made up of ten decades, should begin on a year ending in 00. Why should a new century begin one year after a new decade begins?
And there's no real mathematical sense in insisting that a count has to begin with 1 instead of 0. The ten digits of the decimal system are 0 through 9. Once you surpass 9, you add a number to the tens column and start the ones column over again at 0. So starting with 0 is consistent with the way decimal counting is done.
So yes, there are definitions. But your definitions are wrong and based in tight-assed prescriptivism rather than good sense. They're just an excuse to be smug and elitist and call other people idiots. And that's as gross an abuse of language as what you're accusing others of doing.
No, because there was no year zero. That's not how years are counted in the calendar we use. Maybe when you pick up a dozen donuts, you count them from 0 through 11, but most people start counting with 1. Decades and birthdays are counted differently. The calendar goes from 1BCE to 1CE. Therefore the 1st Century goes from 1CE to 100CE. The 2nd Century goes from 101CE to 200CE. Extrapolating forward, we can see that the 21st century goes from 2001CE to 2101CE. The only reason that people think the 21st century begins in the year 2000 is because they see all those zeros come up like on their car odometer and they don't know any better.