Are there any well-established conventions to distinguish the two senses of being an ultrafilter on a set $X$ (when $X$ happens to be equipped with an ordering)?
This ambiguity is confusing at first; I was confused about the "type mismatch" until I happened to see the section of the Wikipedia article mentioning the ambiguity explicitly.
Let $\Lambda$ be $2^\mathbb{N}$. Let $\Xi$ be $2^{2^\mathbb{N}}$.
I've been studying ultrafilters recently as part of the construction $\prod^U_{i \in \mathbb{N}} M_i$, which takes a sequence of models, takes their Cartesian product, and then mods out by an equivalence relation $\equiv$ that holds if and only if the set of indices of agreement between $t$ and $s$, $\{i : t_i = s_i\}$, is an element of $U$.
This makes sense, and this would make $U$ an element of $\Xi$.
However, $\mathbb{N}$ itself has an ordering that's very natural, namely $\le_\mathbb{N}$. And ultrafilters can be defined on ordered sets directly.
So that made we wonder whether $U$ was an ultrafilter "on" $\mathbb{N}$ or an ultrafilter "on" $\Lambda$.
The Wikipedia article on ultrafilters has the answer, quoted below.
An (ultra)filter on $\wp(X)$ is often just called an (ultra)filter on $X$.
The footnote elaborates.
If $X$ happens to be partially ordered, particular care is needed to understand from the context whether an (ultra)filter on $\wp(X)$ or an (ultra)filter just on $X$ is meant; both kinds of ultrafilters are quite different.
Off the top of my head, I can think of using an ultrafilter directly on $X$ as the basic case when $X$ is an ordered set and an ultrafilter obliquely on $X$ to mean an ultrafilter directly on $2^X$.
Wikipedia also has an example in the same footnote of using of to mean directly on and reserving on to mean obliquely on. I don't know widespread this convention is.
Some authors use "(ultra)filter of a partial ordered set" vs. "on an arbitrary set"; i.e. they write "ultrafilter on $X$" to abbreviate "ultrafilter of $2^X$".