just in case you missed it.. an excellent example you brought up.
from wikipedia...
Chester A. Arthur (1829â1886), 21st president of the United States, was rumored to have been born in Canada.[35][36] This was never demonstrated by his Democratic opponents, although Arthur Hinman, an attorney who had investigated Arthur's family history, raised the objection during his vice-presidential campaign and after the end of his Presidency. Arthur was born in Vermont to a U.S. citizen mother and a father from Ireland, who was eventually naturalized as a U.S. citizen. Despite the fact that his parents took up residence in the United States somewhere between 1822 or 1824,[37] Chester Arthur additionally began to claim between 1870 and 1880[38] that he had been born in 1830, rather than in 1829, which only caused minor confusion and was even used in several publications.[39] Arthur was sworn in as president when President Garfield died after being shot. Since his Irish father William was naturalized 14 years after Chester Arthur's birth,[40] his citizenship status at birth is unclear, because he was born before the 1868 ratification of the 14th Amendment, which provided that any person born on United States territory and being subject to the jurisdiction thereof was considered a born U.S. citizen, and because he was a British subject at birth by patrilineal jus sanguinis.[41] Arthur's natural born citizenship status is therefore equally unclear.