Given any equivalence $\sim$ on $X$ do functions $f:X\to X$ such that $f(u)=f(v)\iff u\sim v$ have a specific name?
I know if I replace the image of each element with its pre-image then this is called a canonical projection map of $\sim$.
Also that this set of functions $S$ forms a sub-monoid $M=(S,\circ,\text{id}_X)$ of the complete transformation monoid on $X$ (the monoid of all functions mapping elements from $X$ to $X$).
Observe that in fact any function $f:X\to Y$ determines an equivalence relation $\sim$ on $X$ as follows: for all $x,x' \in X$,
$$x \sim x' :\Leftrightarrow f(x) = f(x').$$
That implies that any function with $\sim$ given as above has a natural decomposition (*) $$ X\xrightarrow{p} (X/\sim) \xrightarrow{\overline{f}} f(X) \xrightarrow{i} Y$$ such that $f = i\circ \overline{f}\circ p$ (unfortunately i can't draw any commutative diagram via mathjax).
In the diagram above $p$ is the natural projection, $\overline{f}$ is a bijection defined by $\overline{f}([x]) = f(x)$ and $i$ is the inclusion $f(X)\subset Y$.
You'll notice that if $f$ is surjective, the diagram simplifies to
$$ X\xrightarrow{p} (X/\sim) \xrightarrow{\overline{f}} Y.$$
(*) this might be the term you were looking for.