Yep that's right. Jurisdiction applies to people, subject matter and territory. How does that negate my point? Yep, it doesn't.
"You never responded to my hypothetical of children born to parents of foreign nationals on a foreign military campaign on U.S. territory"
I never answered it because it's a hypothetical and irrelevant to the topic of illegal immigrants. They're illegal immigrants, and not an army under arms, carrying a foreign flag. If you believe such an hypothetical application of Ark is feasible, then the argument has devolved into such absurdity, it becomes pointless.
"Under your simplistic analysis, such children would be U.S. citizens because the U.S. asserts jurisdictional rights over the “territory” on which such children were born."
Reductio ad absurdum - look it up.
You still either can't, or don't want to, comprehend that people can only be (your words) in "violation of our laws", if they are indeed "subject to the jurisdiction thereof". If they aren't under US jurisdiction - either territorial, personal or subject matter jurisdiction - then there can be no violation of US law.
Because it can have a different application depending on the object or purpose with respect to which the concept is applied. At the time of adoption of the Fourteenth Amendment senators spoke of the "extent and quality" of the jurisdiction to be applied. Your "pregnancy" analogy is oversimplified as applied to these cirumstances.