A decade is simply a time span of 10 years and a century a span of 100 years. The start dates of each are determined by how they are being used.
The first decade of 21st century is 2000s
I think that statement is where the problem lies. Strictly speaking it isn't true.
Under the Gregorian calendar, the 21st Century started on January 1, 2001. The date is fixed by the fact that there was no year zero. So every century within that calendar started on January 1, XX01.
As far as decades within centuries, it's a matter of how they are being used. Most people would interpret the "2000s" as being January 1, 2000 to December 31, 2009 because they would interpret 2010s to be from January 1, 2010 and ending on December 31, 2019, etc.
However, if you were talking specifically about the first decade of the 21st Century you'd be discussing January 1, 2001 to December 31, 2010.