In the Netherlands at least, we use "familie" to refer to the extended family. To talk about the immediate family or one's household, we would use "gezin" (Although I think some parts of the Netherlands may use "familie" for the nuclear family as well? Not where I live, though.)
We don't generally distinguish between matrilineally and patrilineally related family, which is what the "cousin outside the family" was originally about.
Using one word for your family and an unrelated word for your other relatives, including your cousins, doesn't make it sound like your cousins are part of your family. That's a sharper distinction than is drawn in English.
This isn't a question of the pronunciation of the word you use to refer to remote relatives. It's a question of the social relationships that exist between you and your relatives.
We don't generally distinguish between matrilineally and patrilineally related family, which is what the "cousin outside the family" was originally about.