You are correct that Cruz is not eligible to be President, and you correctly state reasons why according to the 14th ammendment, Cruz was NOT even a U.S. citizen at birth.
To further clarify, the 14th ammendment defines citizen at birth, NOT “Natural Born Citizen” which is not mentioned anywhere in that ammendment.
To understand that, look to Emer de Vattel, who defined a ‘natural-born citizen’ in his highly acclaimed and influential “The Law of Nations, Or, Principles of the Law of Nature, Applied to the Conduct and Affairs of Nations and Sovereigns,” Section 212 (1758 French) (1759 English) and to several Supreme Court cases that cite Vattel and his explanation of “Natural Born Citizen” as one born on U.S. soil to citizen parents, such as:
The Venus, 12 U.S. 8 Cranch 253 253 (1814)
Vattel, who, though not very full to this point, is more explicit and more satisfactory on it than any other whose work has fallen into my hands, says: “The citizens are the members of the civil society; bound to this society by certain duties, and subject to its authority, they equally participate in its advantages. The natives or indigenes are those born in the country of parents who are citizens.
Society not being able to subsist and to perpetuate itself but by the children of the citizens, those children naturally follow the condition of their fathers, and succeed to all their rights.
Minor v. Happersett , 88 U.S. 162 (1875)
The Constitution does not in words say who shall be natural-born citizens. Resort must be had
elsewhere to ascertain that. At common law, with the nomenclature of which the framers of the
Constitution were familiar, it was never doubted that all children born in a country of parents who
were its citizens became themselves, upon their birth, citizens also. These were natives or natural-
born citizens, as distinguished from aliens or foreigners. Some authorities go further and include as
citizens children born within the jurisdiction without reference to the citizenship of their parents. As to
this class there have been doubts, but never as to the first.
United States v. Wong Kim Ark, 169 U.S. 649 (1898)
At common law, with the nomenclature of which the framers of the Constitution were familiar, it was
never doubted that all children, born in a country of parents who were its citizens, became
themselves, upon their birth, citizens also. These were natives, or natural-born citizens, as
distinguished from aliens or foreigners.
The US Constitution gives Congress the authority to make laws governing immigration and naturalization, they have no authority over nature and those born naturally to the members of any given society.
Citizenship is a political characteristic that can only come via 2 ways, being born to citizen parents that increases the society by natural means, or by naturalization, that which increases the membership of a society by the rule of law.
Rogers v. Bellei (1971) https://supreme.justia.com/cases/federal/us/401/815/case.html