How do we know that this statement is correct? Is it in the Constitution? (No). Could this language be in a statute or in naturalization regulations? I don't know, but I haven't seen it. I have seen it repeatedly on Free Republic, but never with a source.
This statement is not correct. The source typically used to create a distinction is Blackstone, but if you read chapter 10 you will find that the terms are used interchangeably. The other sources used, such as Vattel, don't even agree on a definition. It's basically grasping at straws.