UCLA has a journal of heritage language learning at
http://www.international.ucla.edu/languages/heritagelanguages/journal/
Heritage Language learners often refer to students whose parents are either immigrants from another culture or are members of a minority language community. These students may know how to speak their heritage language (or perhaps familiar with only a few phrases), but have never been formally trained in educated “grammar” (including spelling or writing).
Heritage learners usually don’t need to start from zero, but typically do need information on what the educated standards, information on technical vocabulary and exposure to the culture’s literary classics (in the actual language).
Depending on the situation, a heritage learner may be stigmatized by the educated speakers of the language as “not really speaking correctly”.
Note, there may also be learners whose ethnic background is associated with a certain language, but are at the same stage as true beginners.