[Ready4Freddy]

“and, thereby, became a naturalized citizen of the United States”

That’s where you went off track; Cruz was a US citizen at birth by virtue of his mother’s US citizenship.

[RC one]

He may have been a citizen but he certainly wasn't a natural born citizen. If anything, he is a citizen of the United States and a Natural Born Citizen of Canada by virtue of his father's nationality and geographic location. When the NBC clause was adopted, it was widely understood that a child born on foreign soil to a foreign father would "follow the condition of the father"; in other words, inherit the father's nationality.

This is made clear in Vattel's Law of Nations, Book I, Ch. XIX:

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 natural-born citizens, are those born in the country, of parents who are citizens. As the society cannot exist and perpetuate itself otherwise than by the children of the citizens, those children naturally follow the condition of their fathers, and succeed to [218] all their rights. The society is supposed to desire this, in consequence of what it owes to its own preservation; and it is presumed, as matter of course, that each citizen, on entering into society, reserves to his children the right of becoming members of it. The country of the fathers is therefore that of the children; and these become true citizens merely by their tacit consent. We shall soon see, whether, on their coming to the years of discretion, they may renounce their right, and what they owe to the society in which they were born. I say, that, in order to be of the country, it is necessary that a person be born of a father who is a citizen; for if he is born there of a foreigner, it will be only the place of his birth, and not his country.

And

It is asked, whether the children born of citizens in a foreign country are citizens? The laws have decided this question in several countries, and their regulations must be followed. By the law of nature alone, children follow the condition of their fathers, and enter into all their rights (§212); the place of birth produces no change in this particular, and cannot of itself furnish any reason for taking from a child what nature has given him; I say of itself, for civil or political laws may, for particular reasons, ordain otherwise. But I suppose that the father has not entirely quitted his country in order to settle elsewhere. If he has fixed his abode in a foreign country, he is become a member of another society, at least as a perpetual inhabitant; and his children will be members of it also.