Nope. What you are missing is that "natural-born citizen" had a specific meaning in common law. Most of the Founders were lawyers. They knew exactly what it meant. There was no need to define it in the Constitution, as it already had a specific meaning.
Don't take my word for it. Read the Wong Kim Supreme Court decision. The Court explains what these terms mean and their history in tens of pages of detail.
Don't argue with me. Argue with the Supreme Court.
For easy reference, there’s a wiki article about that ruling with links:
http://en.wikipedia.org/wiki/United_States_v._Wong_Kim_Ark