Repeating a non-sequitor doesn’t help the discussion.
Let’s consider:
1) “All Xs are Ys”... does that mean only Xs can be Ys? of course not. So stop presuming that a quote from a man saying “All Xs are Ys” is telling us than any non-X cannot be a Y. Fallacious. We know from Wong Kim Ark that the definition of birthright citizenship was changed by the 14th amendment to exend beyond parents who are citizens.
2) Your second logical fallacy is presuming that a definition of natural-born citizen is fixed by these quotes while ‘citizen at birth’ can change. In other words, ‘natural born’ I have argued is a term of art that equates to citizen at birth.
3) The third error is presuming that a quote from a Congressman is controlling in any manner. It’s better than a foreigner, but it’s still not legally binding. Bingham was not even talking about the 14th Amendment, and the language from that Act was different from what was put into 14thA. So it’s merely descriptive of his view of citizenship and no more so than a number of other quotes, from court rulings, etc.
So you have a non-controlling opinion from a politician about a law that it not the current law, stating a category of people who are natural-born but not addressing in any way the question at hand of whether there is a distinction or a class of people who are citizens at birth but NOT ‘natural-born citizens’. Nothing he says lend support to that assertion.
Again, “natural-born citizen’ and ‘citizen at birth’ are synonymous. There is no distinction in US law that separates the two, and nothing you have brought here changes that basic and simple point.
I found this interesting. From Joseph Story, Commentaries on the Constitution 3:§§ 1473 , 1833.
§ 1473. It is indispensable, too, that the president should be a natural born citizen of the United States; or a citizen at the adoption of the constitution, and for fourteen years before his election. This permission of a naturalized citizen to become president is an exception from the great fundamental policy of all governments, to exclude foreign influence from their executive councils and duties. ...
So Story seems to indicate that the eligibility clause is there to prevent naturalized citizens from becoming president, with the exception of those who were naturalized citizen at the adoption of the constitution. In his commentary, he doesn't make a distinction based on the parent's status.
I don't know how you prove that a person who at birth becomes a citizen is a naturalized citizen.