No, that is exactly the opposite of what Vattel says. Vattel does link citizenship to allegiance and jurisdiction.
The first immigration act was passed by our First Congress in 1790. In chapter III is Vattel’s assertion that citizenship is derived from the father, in that citizenship was prohibited to children whose fathers have never gave intent to permanently reside of the Untied States.
Interestingly in this same act, we also find the clarification of a Natural Born Citizen, as being one “And the children of citizens of the United States, that may be born beyond sea, or out of the limits of the United States, shall be considered as natural born citizens: Provided, That the right of citizenship shall not descend to persons whose fathers have never been a resident in the United States:”
Residency was defined in that same act as someone under oath declaring that they wished to remain and live in the Untied States.
Cruz was born under the jurisdiction of Canada. His father was under the jurisdiction of Canada. Because his mother was most likely an American citizen at the time he was entitled to become an American citizen by statute but he was not born an American citizen, it had to be adjudged. Furthermore his father had not fulfilled the condition of residency.
Cuba's law made Ted a citizen of Cuba at birth. Ted was born the citizen of three countries. Canada by birth, US by statute, and Cuba by statute.