Yes, Ted Cruz is a "natural-born" citizen -- even in an originalist reading of the clause

This is specious argument. The 1790 Act reveals that the members of Congress–many of whom were heavily involved in the writing and ratification of the Constitution–understood that children of U.S. citizens who were born abroad should be “considered” as “natural born” in the sense that they did not need to undergo any naturalization process and were accordingly legally entitled to be considered U.S. citizens at the time of their birth–the same as an individual born within U.S. borders. The fact that Congress memorialized this common understanding in the 1790 Act does not, in any way, suggest that such children born abroad “had to be naturalized”; quite the contrary.

In short, while Trump and Harvard Law prof Laurence Tribe are correct that the U.S. Supreme Court has not definitively grappled with the full meaning of “natural born citizen,” the available evidence suggests that if/when the Court ultimately must grapple with it, the evidence points strongly in Cruz’s favor.