Your definition of natural born citizen has no support in case law, nor has any contextual basis in the Constitution.
In fact, there’s many been numerous Presidents who don’t meet your criteria of “natural born citizen.”
Other than Arthur, can you name other American Presidents who did not meet the definition?
The criteria is were both parents citizens of America at the time of the birth of the child.
The definition of the term, “natural born citizen”, was entered into the Congressional record of the House on March 9 in comments made by Rep. John Bingham on the Civil Rights Act of 1866, which was the precursor to the Fourteenth Amendment. He repeated Vattel’s definition when he said:
[I] find no fault with the introductory clause, which is simply declaratory of what is written in the Constitution, that every human being born within the jurisdiction of the United States of parents not owing allegiance to any foreign sovereignty is, in the language of your Constitution itself, a natural born citizen.
John A. Bingham - March 9, 1866 Cong. Globe, 39th, 1st Sess., (1866), page 1291
[Sorry for the cut & paste, but the Library of Congress disables active links]
http://memory.loc.gov/cgi-bin/ampage?collId=llcg&fileName=071/llcg071.db&recNum=332