Both on Wikipedia and on Merriam-Webster, "family name" is given as a synonym of "last name" or "surname". In my own language as well, "family name" is simply u

Both on Wikipedia and on Merriam-Webster, "family name" is given as a synonym of "last name" or "surname". In my own language as well, "family name" is simply understood as a part of your legal name with slightly different rules. Perhaps in genealogy circles your definition applies. But it's certainly not how the word (well, expression?) is used in common language in English, and I bet in other languages either.