Since you think Cruz was a natural born Canadian as opposed to being both a natural born Canadian and a natural born American (and now having renounced his Canadian citizenship, is a person of no citizenship), I have a question:
Donald Trump and Ted Cruz propose that the United States practice of considering children born in the United States to foreign citizen parents who are here on temporary visa or who are here illegal NOT be natural born U.S. citizens. (Possibly, the President could simply direct that the practice of recognizing them as such be ended. Possibly this would require an act of Congress or even a Constitutional Amendment.)
If we were to make this change, to what country would persons born here to such parents be citizens?
Before I answer you question I want to address the natural born Canadian birth of Sen.Ted Cruz. One way to renounce U.S.citizenship is by voting in another nation.
The Vietnam War was ongoing at the time Rafael Cruz and Eleanor Wilson moved to Canada. Neither supported the U.S. activities in southeast Asia and both voted in Canadian election(s). When little Teddy Cruz was born in 1970 his birth was not registered with the U.S. Embassy. He was Canadian because his parents were recognized as Canadian.
Dual citizenship was not recognized by Canada until 1976/1977. So when Teddy left Canada with his mother in 1974, the only way he could have entered the U.S. was on his mothers Canadian passport.
It is up to Sen.Cruz to fix his citizenship. A person born in another nation is not eligible to be President.
Children born in the United States to foreigners can be registered with the U.S. government or registered with the Embassy of the parents nation.
Some foreign governments do not recognize **birthright**; Mexico does not, so a child born in the USA to one or more Mexican citizens are Mexican citizens themselves.
Mitt Romneys father, George Romney, was born in Mexico but was a U.S. citizen due to the Mexican laws about citizenship. Being born in Mexico does not equal citizenship.
To answer your question, just because a child is born in the USA does not automatically mean they are citizens. If the parents are long term resident aliens, then yes, the baby can be a citizen. If the parents are here temporarily OR come from a nation that does not recognize birthright (such as Mexico), then too bad, so sad, the baby is not a U.S. citizen. These are how our laws exist now. They are not being enforced.
Let me give you an example. A good friend of mine was born in Chihuahua, Mexico, while her mother was visiting relatives. Her father was a U.S. citizen and the parents residence was in Los Angele, CA. My friend was a Mexican citizen because her mother had not yet naturalized as a U.S. citizen. When she was 18 years old she naturalized as a U.S. citizen herself. Do you see how the existing laws work? I do not believe we need to alter the laws.