While I stand by your desire for more nuance in the discussion, I'd be wary of assuming one theorization of happiness, and it's attendant conceptual divisions, is the only theory of happiness. While the hedonistic and eudaemonic conceptions come down to us from tradition for good reason, it seems somewhat presumptuous to call these categorizations the "proper names" for the kinds of happiness.

Since it's a humanistic concept, there are many ways to break it down and define it--some competing theories might not even agree with the hedonistic/eudaemonic split (some might find the division too simple, others unnecessary).

There's a lot of different ways one can approach it. For instance, the discussion gets even more interesting when we consider, on the hedonistic side, the world of Sadism--as concepts that seems diametrically opposed (pleasure/pain) intermingle quite interestingly. I would not be surprised if a similar phenomena occurs on the level of hedonism/eudaemonia--the pursuit of one may also mix with some of the incentives and pleasures of the pursuit of the other.