No, if they were the same thing the Founders wouldn't have used the term Natural born as a requirement for Presidential eligibility. They would have used the same term as they did for all of the others - citizen of the United States.
I keep asking to be shown sources where more than two categories of citizenship are cited.
I never said there were more than two.
If your citizenship status has to be determined by a law, you're naturalized, not natural born.
And no, I won't sort through and define anything to your satisfaction. The burden of proof is for you to come up with a law that uses the term 'natural born' since that's what the Constitution requires.
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BTW - Wong Kim Ark, the benchmark derision that supposedly created a 'native' born citizen when someone is born here of foreign parents said-
A person born out of the jurisdiction of the United States can only become a citizen by being naturalized, either by treaty, as in the case of the annexation of foreign territory, or by authority of congress, exercised either by declaring certain classes of persons to be citizens, as in the enactments conferring citizenship upon foreign-born children of citizens,..
With few exceptions, being born in a another country means you're born outside the jurisdiction of the United States.
In order to be considered a "naturalized" citizen, you must have undergone the following. There is no other way to be considered naturalized. If you can produce information that Ted Cruz went through this process, I will concede your point. If he was not a citizen at birth and did not undergo the naturalization process, then he has been illegally serving as a U.S. Senator.
Naturalization Process
In general, the naturalization process includes the following steps:
Determine if you are already a U.S. citizen.
Determine your eligibility to become a U.S. citizen.
Prepare Form N-400, Application for Naturalization.
Submit Form N-400, Application for Naturalization.
Go to the biometrics (fingerprinting) appointment, if applicable.
Complete the interview.
Receive a decision from USCIS on your Form N-400, Application for Naturalization.
Receive a notice to take the Oath of Allegiance.
Take the Oath of Allegiance to the United States.
Understand your rights and responsibilities as a U.S. citizen.