What is the smallest digraph whose reflexive, symmetric, transitive closures (in all combinations) are distinct?

The $4$-vertex digraph

a ---> b ---> c    d

is the smallest example possible.

To have the reflexive symmetric transitive closure be different from the symmetric transitive closure, we need an isolated vertex. (If a vertex $v$ has an edge to or from it, then in the symmetric transitive closure, we get the edge $v \to v$.) That isolated vertex will make all the reflexive closures different from the non-reflexive ones, but can't help us with anything else.

For the digraph a ---> b ---> c we can check that symmetric, transitive, and symmetric transitive closures are all different. If we want to beat this, we need the same thing to happen on a $2$-vertex digraph.

If the $2$-vertex digraph has edges $a \to b$ and $b \to a$, then its symmetric closure will not change anything. However, if the $2$-vertex digraph does not have both of those edges, then its transitive closure will not change anything. So either way, we need $3$ vertices.