Semidegree two in a least-order counterexample

Let $V$ be a type in a fixed universe with a finite-type structure, let $G$ be a simple graph on $V$, and let $D$ be a binary relation on $V$. Assume that $G$ admits an injective straight-line embedding in the plane in which no vertex lies on an edge not incident with it and any two edges whose endpoints are pairwise distinct have disjoint closed segments. Assume that $D$ has no loops, every arc of $D$ follows an edge of $G$, and every edge of $G$ has exactly one of its two directions in $D$. Assume additionally that no coloring $c:V\to\mathrm{Fin}\,2$ makes both color classes free of directed cycles, where a directed cycle is a list $x$ followed by a middle list and then $y$, of length at least $3$, with no repeated vertices, directed arcs between consecutive entries, and an arc from $y$ back to $x$; also assume that every finite oriented planar graph on a type $W$ in the same universe with $|W|<|V|$ does admit such a coloring. Then, for every vertex $v\in V$, the set of in-neighbors $\{u\in V\mid D(u,v)\}$ has cardinality at least $2$, and the set of out-neighbors $\{u\in V\mid D(v,u)\}$ has cardinality at least $2$.