The common factor is - just as stated by James Bayard and approved by Chief Justice John Marshall - citizenship by birth.
There are two routes to citizenship by birth. Jus soli is the primary one, jus sanguinis is the secondary one. Both are pathways to natural born citizenship.
It's not totally official in the case of the second route (though it is in the case of the first), but I'm sure the Supreme Court would approve a Ted Cruz candidacy.
Cruz is not a “citizen by birth”, he is by US law a “citizen at birth” by naturalization statute.
Without naturalization statute would Cruz be a legal US citizen? If “yes” then what is the necessity of the statute? The answer must be “no”.