Ambiguity of the term is no excuse.

Indeed. I plead guilty to reading hastily. I saw the term "meet" being used in a context where I already knew its definition (the only definition it had, so far as I knew), so I only briefly skimmed Wei Dai's own definition. Obviously I was too careless.

However, it really bears emphasizing how strange it is to put refinements higher in the partial order of partitions, at least from the perspective of the general theory of partial orders. Under the category theoretic definition of partial orders, P ≤ Q means that there is a map P → Q. Now, to say that a partition Q is a coarsening of a partition P is to say that Q is a quotient P/~ of P. But such a quotient corresponds canonically to a map P → Q sending each element p of P to the equivalence class in Q containing p. Indeed, Wei Dai is invoking just such maps when he writes "I(w)". In this case, Ω is construed as the discrete partition of itself (where each element is in its own equivalence class) and I is used (as an abuse of notation) for the canonical map of partitions I: Ω → I. The upshot is that one of these canonical partition maps P → Q exists if and only if Q is a coarsening of P. Therefore, that is what P ≤ Q should mean. In the context of the general theory of partial orders, coarser partitions should be greater than finer ones.