In the same manuscript, I'm using unary upper closure operators, which are idempotent, and binary lattice join operators, which are also idempotent.
An unary operator $f$ is idempotent iff $f{\circ}f=f$.
A binary operator $\vee$ is idempotent iff $x{\vee}x=x$ for all $x$ from the underlying lattice.
While it is ok to have this little amount of ambiguity in the definition of idempotent, it would be nice to avoid ambiguity completely. Is there any established synonym for any of the two kinds of idempotence?
The difference is that in the first case, you consider an idempotent element and in the second case, an idempotent semigroup$^{(1)}$. More precisely
Definition. Let $(S,*)$ be a semigroup. An element $e$ of $S$ is idempotent if $e * e = e$. Moreover $S$ is idempotent if every element of $S$ is idempotent.
Now $f \circ f = f$ means that $f$ is idempotent in the semigroup of unary operators under composition. When you say that a binary operator $\vee$ on a set $X$ is idempotent, it means that the semigroup $(S, \vee)$ is idempotent.
$\scriptsize\text{(1) I assume that your operator $\vee$ is associative. If it is not the case change "semigroup" to "magma" in the definition.}$