What do people usually mean when they ask "Do you have a family?"
When American people ask you, "Do you have a family?" does that mean "Do you have any children?" or just "Are you married?"
The question is susceptible to a broad interpretation. It can mean either of the two meanings you suggest, or it can be an inquiry about parents, brothers and sisters and cousins and possibly others. The circumstances of the question will usually make it clear what is meant.