Non-adjacent triples of cells in a triangular grid: proofs of three conjectured generating functions

Let $\Tn$ be the triangular grid of side $n$, with its $n^2$ unit cells, and let two cells be side-adjacent if they share an edge and point-adjacent if they share a vertex. We count the orbits of $3$-element sets of pairwise non-adjacent cells under the rotation group $C_3$ and under the full symmet