“These were natives, or natural-born citizens, as distinguished from aliens or foreigners.”
Was Cruz an alien or foreigner at birth? No. He was born a citizen.
One is either born a citizen with natural allegiance, or one is an alien or foreigner and must be ‘naturalized’ into a condition of allegiance.
There's no reason to gloss over this. Vattel said that some countries naturalize aliens at birth. Such falls under a category of collective naturalization. Cruz's citizenship can only be established under statutory U.S. law, which means that under natural law, he is definitely considered to be an alien at birth.