He has less than NO reason to lie about this, Obama wasn't running for president, Cruz wasn't even on anyone's radar, the government had long ago okayed McCain's NBC, so it was just something that tickled him, because his wife was preggars at the time and he knew that he would be an American citizen when his wife delivered their baby.
It wasn’t that hard. I was able to pull up the study guide and found the Citizenship pdf. It is all there, but the in the title of the text definition(s) is that icky word, “generally”.
No wonder historians say the thing is “unsettled”.
It was a little deep for me, but I’m tired. You could probably enjoy the read. It isn’t long. I googled naturalization study guide and was able to search in the guide for citizenship requirements.