In the social sciences and in day-to-day politics we often operate with the concept of immigrant background. It’s a loose concept, and often used without adequate consideration. In the social sciences, we often define anyone who has at least one foreign parent as having an immigrant background. This is systematic, but not the solution.
Following this approach all “mixed” children of one native parent and one foreign parent are considered “foreign”. In some way, that’s akin the one drop of blood used to define what counts as black in the US; a reflection of concerns over purity and boundary-making rather than attempts to create an empirically useful category.
We should simply get away from the idea that a single definition fits all our concerns. Mixed children are native speakers and are fully part of the local culture. The fact that one of their parents is “foreign” is not a deficit, but something they have in addition. This is a different case from having two foreign parents where something (language, attitudes, culture) might be missing. In a different situation, however, having a single foreign parent might be equivalent to having two. For instance, if we’re looking at discrimination, one foreign parent might be a relevant negative marker making the individual more susceptible to discrimination.