@Andi: Yes. I see that Alexandra's categories listed as the four members of the supercategory Aufenthaltsgenehmigung (v. http://www.aufenthaltstitel.de/stichwort/aufe...
), and that the first reference on that page ( http://www.aufenthaltstitel.de/auslg.html#5
) points to the legislative definitions of the different conditions applicable to each. (Later links point to summaries of each.)
@Alexandra: I suspect you'll be able to summarise your own descriptions from those. I'd be happy to suggest any refinements to them, if needed.
What I initially found confusing is that the four category names each, to my limited knowledge, seem to mean much the same thing. It seems to me now that I'd translate Aufenthaltsgenehmigung as 'leave to remain'. In the UK ontology, that can then be divided into conditional (eg as to length, or purpose, or activities allowed or renewability) and unconditional (undefinite).
There are then various categories of immigrant - visitor, student, work permit holder, EEA (?EU - what _is_ it called now?) worker, self-employed, artist, refugee or asylum seeker, etc, etc, or the spouse or dependant of any of these - and then rules as to which kind of leave may be granted, and when, to applicants in each of these categories. It looks to me, from a very brief scan, as if your four categories are each baskets for different blends of type of leave and applicant.