“you seem to miss the distinction”
It doesn’t exist. At least not since the passage of the 14th amendment.
The primary author of the citizenship clause, Sen. Jacob M. Howard, said the “word jurisdiction, as here employed, ought to be construed so as to imply a full and complete jurisdiction on the part of the United States, coextensive in all respects with the constitutional power of the United States, whether exercised by Congress, by the executive, or by the judicial department; that is to say, the same jurisdiction in extent and quality as applies to every citizen of the United States now.”
United States Attorney General, George Williams, whom was a U.S. Senator aligned with Radical Republicans during the drafting of the Fourteenth Amendment in 1866, ruled in 1873 the word “jurisdiction” under the Fourteenth Amendment “must be understood to mean absolute and complete jurisdiction, such as the United States had over its citizens before the adoption of this amendment.” He added, “Political and military rights and duties” do not pertain to anyone else.
Essentially then, “subject to the jurisdiction thereof” means the same jurisdiction the United States exercises over its own citizens, i.e., only citizens of the United States come within its operation since citizens of the United States do not owe allegiance to some other nation at the same time they do the United States.
This makes a great deal of sense for the time because there was a great deal of controversy over conflicts arising from double allegiances.
In fact, Congress issued a joint congressional report on June 22, 1874 that said the “United States have not recognized a double allegiance.”
http://federalistblog.us/2008/11/natural-born_citizen_defined.html