Ingeborg gave us the right link in #10, the definition of those who are
not Herkunftsdeutsche is
1. zugewanderte und nicht zugewanderte Ausländer;
2. zugewanderte und nicht zugewanderte Eingebürgerte;
3. (Spät-)Aussiedler;
4. mit deutscher Staatsangehörigkeit geborene Nachkommen der drei zuvor genannten Gruppen.
That is, like the term Migrationshintergrund it has nothing to do with ethnicity or nationality. Pipper's indigenous in #23 is a corresponding term, defined as "not introduced directly or indirectly ... into a particular land or region" (M-W).
Biographies such as those in #7 and #21 were the challenges of a previous era of social integration, in which the boundaries of "Germanness" were fluid, and contested though ideology and war.
I do not think the clumsy expression "Menschen mit Migrationshintergrund" should be criticized (O-Ton "verzweifelter Versuch") for not addressing the complexities of previous generations' biographies, which as they graphically describe were complex and challenging. The term addresses contemporary issues which are bound up with migration in a context of globalisation.
It is preferable to one referring to ethnicity, because the disadvantage that people from "elsewhere" experience, and the particular hurdles they have to overcome, have their origin in this "elsewhere", language, education, culture, irrespective of their skin colour or other marker of ethnicity. The term in this way can include those migrants of German origin who, before, in post-WW2 policies, were privileged for their ethnicity. This in itself is quite a step forward for German social policy.
Most of the everyday equivalents in AE and BE will not manage to get away from the "ethnic" perception of minorities in social policy (cf #23, "white British" etc.), understandable in the context of the USA and in the history of UK migration from the Commonwealth. But that would misrepresent the German concept.
