Upvote:2
October 4 to October 19 is indeed sixteen days as far as visa regulations are concerned. You have to count each calendar day during which you will be in the Schengen area, including the days at each end of your stay and not simply the length of time.
If you are not able to come on October 5 instead, the best would be to contact the consulate and ask them what to do. They can certainly amend the visa, especially if they made a mistake (which does not appear to be the case here however). Beside the cost, applying for a new visa without clearing that with them beforehand is likely to cause even more confusion.