In short, you subscribe to the definition of the Mr. Waite that you reference.
However, most reasonable people would not, instead recognizing that “natural born citizen” means exactly what it says: a citizen born on the United States, regardless of their parents’ status.
You are not a reasonable person in this instance, because you refuse to recognize with reason how you are using the terminology exactly contrary to its meaning and historical usage. Citizenship by the doctrine of jus soli is a form of statutory citizenship granted to a person by the act of a statute. Consequently, citizenship by statutory act of jus soli (law of the soil) cannot be its very definition be its opposite, natural born citizenship arising out of natural law and not statutory law. Only jus sanguinis (law of the blood) arises from the law of nature, natural law, and natural born citizenship. This was acknowledged in a U.S. Supreme Court case by the explicit description of how a nation is embodied under natural law by the people born with blood relationships to the parents of that nation.
Furthermore, your interpretation of the clause is nonsensical, because it fails to note how the Constitution used the singular word of citizen in one place and the more narrowly defined natural born citizen clause in the other part of the Constitution. You cannot have it both ways by trying to wrongfully conflate citizen to mean the same thing as natural born citizen. They were different in 1797, and they remain different in 2012.