There is no "0" year, 1 is the 1st year, so 100th year is still the 1st century, therefore 2nd century starts in 101 and 20th in 1901.
We have special rules for leap years, that would just be a single leap-back century.
At the scale of centuries, starting 2nd century at 100 as opposed to 101 is just an 1% error, so we can live with it. For the kind of uses we use centuries for (not to do math, but to talk roughly about historical eras) it's inconsequential anyway.