It's not that Nicolas = Santa Claus in Dutch, but it's like this. His name originated as Saint Nicholas - and over time, gradually became Santa Claus.
This is what the site says:
"He is also known as Santa Claus (from Dutch Sinterklaas), the bringer of Christmas presents."
Yes, in some countries, the giving away of presents is celebrated on a different day - such as December 5th, as you pointed out. Since Christmas is really supposed to be a day commemorating the DEATH of Christ (Christ's mass [death]), but since Easter has become the day of His death and resurrection, Christmas has become a celebration of His birth. So when gifts are given are entirely up to the customs of your country. Just wondering, is December 5th the day you celebrate the visit of the Magi (wise men), or do you do that on January 6th? I need to brush up on my Netherlands culture again since most of it seems to have gone away...