You have hit the nail on the head. This is the distinction that many people wish to ignore, citizen by nature vs. citizen at birth by law. But for this argument to prevail there must be the acceptance of the notion of natural law, which unfortunately I doubt exists today. Natural law would restrict marriage to a man and a woman. But we see what the court has done with that. In the end, natural law rests on the belief in God who is the giver of that law. As a society, however, we no longer believe that.
Natural law is now another inconclusive theory in philosophy. It is no longer legal theory.
http://powderedwigsociety.com/eligibility-of-cruz-and-rubio/#
Pretty straightforward and easy to understand presentation about Cruz and Rubio’s eligibility to run for POTUS.