All other names are generally considered either common or historic. Common names are regarded as too ambiguous for scientific use, they are generally only mentioned in relevance to collections such as "How do the local people in <area x> having <population y> of <latin name z> (who might help identify where it is growing) refer to the organism?". In a small number of cases local names confer ethnobotanical or cultural semantics.
I am well aware that laypeople don't always distinguish between various similar species of plants and animals, and I probably can't in some cases myself, but I am specifically interested in some of those "common or historic names" along with their "ethnobotanical or cultural semantics", to see how they might compare with words elsewhere.