Science, biology, and genetics don't always make up the rules when it comes to language used in real world situations
But by that same argument, we cannot claim that the language we use here has any basis in science, biology, or genetics (and we know for a fact that any talk about "races" doesn't). You wanted to have an honest debate about race; part of that honesty is acknowledging that the topic under discussion is largely one of politics, prejudice, and ingrained cultural habits.
Sounds like your wife's family is biracial. Your example does not refute my point.
Literally every single person in America, no matter their skin color, is of African ancestry, and shares physical traits with their African ancestors. Your original definition masks the cultural and political purpose behind racial categories: They exist in order to separate people into specific groups, social shorthands, based on nothing but physical appearance and widely-held prejudice regarding the traits such an appearance might inform. Race is, therefore, an intrinsically political category, informed primarily by political tradition and cultural habit.