Here's a quote from an Assistant US Attorney General, Walter Dellinger, in testimony before members of Congress, which you might find helpful:
"The phrase "subject to the jurisdiction thereof" was meant to reflect the existing common law exception for discrete sets of persons who were deemed subject to a foreign sovereign and immune from U.S. laws, principally children born in the United States of foreign diplomats, with the single additional exception of children of members of Indian tribes. Apart from these extremely limited exceptions, there can be no question that children born in the United States of aliens are subject to the full jurisdiction of the United States. And, as consistently recognized by courts and Attorneys General for over a century, most notably by the Supreme Court in United States v. Wong Kim Ark, (6) there is no question that they possess constitutional citizenship under the Fourteenth Amendment."
Here's the link: http://www.justice.gov/olc/deny.tes.31.htm
'Subject to a foreign power'- a naturalized US citizen may renounce citizenship and take up the citizenship of another country. Hence a person who was born a US citizen may be subject to the foreign power.
Whether someone is a '[British] subject' is also subject to a foreign power is a matter to take up a Queen's Counsel.
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