Wrong. It is not enough to be a US citizen. You must be a natural born US citizen. If you are born overseas of ONE US citizen parent, you are automatically eligible for a green card and a path to citizenship, BUT YOU ARE NOT AUTOMATICALLY a citizen, and you are not a natural-born citizen.
I have friends who were born overseas of a US citizen mother and a Canadian citizen father. They had to file naturalization papers (easy, yes, automatic, no) when they came of age.
I don’t know about that. I am an American citizen and my wife is Japanese, and my daughter was born in Japan. She received CITIZENSHIP (albeit dual Japanese/American) and has an American overseas birth certificate.