Well as with so many things, Thematic and Reference maps can get a bit "fuzzy" around the edges. For one thing, many Thematic maps have a Reference map as a "base map" underneath.

Dasymetic maps take a bit of thinking you wrap your head around. They are sort of recursive in that you are defining regions to aggregate data based on the data you are aggregating. That's what makes them difficult and computationally intensive to produce, but it's also what makes them so good at representing a dataset.

A biome map is more of an abstract categorization based on a wide range of features rather than an aggregation of a numerical data set which is what a Dasymetric map is doing. Think of a dasymetric map as being like a census map where you redefine the census boundaries to try to minimize the variation in population density within a region, for that specific census after gathering the data.

A categorization map is certainly be a thematic map, but they are comparatively simple and so don't have a fancy name. Not that it's easy to make a good categorization map if you have a large and complex set of categories but it's not so much a technical problem as a design one. Geological maps for instance are infamous for their dense categorizations and arcane symbology.