Edit in light of David Ben-Zvi's answer: I was not quite speaking about this trichotomy, but rather mentionning the existence of elliptic categories leading to representations of the elliptic braid group, following the suggestion of David Roberts that one should view quantum groups by their braided monoidal category of modules. But the fact that $r$-matrices and the (dynamical) Yang Baxter equation comes into play is not really a coincidence.
Roughlys speaking, in the KZ world, rational $r$-matrices leads to monodromy representations of the braid group $B_n$, which can be used to construct braided monoidal categories. If you take the trivial $r$-matrice $r=0$ these representations factor through representations of $S_n$, so in some sense you recover the symmetric categories of module over a Lie group. For non trivial rational $r$-matrices, you get braided monoidal categories, and quantum groups by the Kohno--Drinfeld theorem. For trigonometric one, you get representations of the cylinder braid group, the associated categorical notion being that of braided module categorie (hence comodules over quantum groups). Finally, for elliptic solutions of the (dynamical) Yang-Baxter equation, you get elliptic categories. The associated algebraic notion seems to be strongly related with quantum analogs of the algebra of differential operators on a Lie group : http://arxiv.org/abs/0805.2766v2
I hop it makes my answer more clear :)