As a matter of logic, I don't think that holds without adding some limiting assumptions.
There are billions of people outside of the set of "citizen by operation of statute," and by far most of them are not natural born citizens of the US.
To protect Cruz, your position MUST be that "citizen at birth by operation of statute" is the same as natural born citizen. IOW, Cruz is in the set of people made citizen by a naturalization act that includes citizenship by dint of criteria in place at birth.
Birth to U.S. Citizen Parents ("Acquisition")U.S. Citizenship by Birth or Through Parents | Nolo.comIn many circumstances, even though a child is born outside the United States, if at least one parent was a U.S. citizen at the time of the child's birth, the child automatically "acquires" citizenship. When this child marries and has children, those children may also acquire U.S. citizenship at birth.
The laws governing whether or not a child born outside of the United States acquires U.S. citizenship from parents have changed several times. You'll need to look at the law that was in effect on the date of the child's birth (and the parents' birth, if grandparents were U.S. citizens) for guidance. These laws differ for the following time periods:
- prior to May 24, 1934
- May 25, 1934 to January 12, 1941
- January 13, 1941 to December 23, 1952
- December 24, 1952 to November 13, 1986, and
- November 14, 1986 to present.
If having one’s case appear in a statute is “citizen by operation of statute”, then all of us are such, even those born in the USA to 2 citizen parents. ALL appear in the law.
So, the advantage of those born in the USA is really the 14th amendment that specifically says that anyone born in the USA is a citizen. (Even it doesn’t use the term ‘natural born citizen’.)
“Natural born citizen” is no place defined in US law with the only exception being the Naturalization Law of 1790.