>> “I think it means that the writers of the Constitution used vague language in writing the rules for who could become President by using a term that did not get defined in the document.” <<
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A truly absurd statement for a number of reasons.
First, none of the terms used in the constitution are defined in its text, because they were terms of common usage. That is especially true for Natural Born, since the Supreme court had little difficulty coming up with the correct definition four times in succession.
Second, it is rare for a legal document to include definitions of terms unless those terms are newly created for the purpose of the document.
Natural Born Citizen was not a term in common usage in a way similar to 35 or resident. The fact that James Madison thought that birth was the criterion of allegiance, and that to him, place was more important than lineage, shows that some of the founders did not agree with what you “know” is true. So if there were disagreements about the term, and the Supreme Court ruled in Inglis vs Trustees of Sailors Snug Harbor that children of aliens, who are protected by the jurisdiction of the US, owe allegiance and are citizens at birth.
As for your cases, you have it backwards at least in Wong Kim Ark. I have not studied the others, but Im sure you are taking great liberties to try and find a proof for your thesis.