Frequently however, the law and "logic" are two disparate characters in the same play.
You make the same argument many people make here. Then, you go on to quote the Constitution or Vattell or someone else. As a legal argument, it's wholly unpersuasive. If you can point to a precedential Court ruling on the central legal argument - that anchor babies aren't natural-born citizens, then that would be persuasive. You can't.
Until the Supreme Court says that "anchor babies" aren't natural-born citizens, then as a practical legal matter, they are. Moreover, Barack Obama is not an "anchor baby". He is quite clearly a man born on US soil to at least on US citizen parent.
You in turned did not address the issue of why the founders and others use the terms "natural born citizen" and "citizen" distinctly if the two are interchangeable as you seem to assert.