To expand on Allen's answer: the isomorphism type of such a representation is determined just by the images of the incoming maps, which must each be a line if the rep is indecomposable, up to the action of linear transformations on the 2 dimensional space.
the moduli space in indecomposable representations in the former case (ignoring automorphisms) is a single point: any 3 linearly independent 1-d subspaces in a 2 dimensional space (i.e. any 3 distinct points in $\mathbb{P}^1$) can be sent to any other by a linear transformation. This linear transformation is actually unique up to scalar multiplication.
the moduli space in the latter case is $\mathbb{P}^1$ minus 3 points, and the isomorphism is cross ratio: you use the isomorphism above to send 3 of the lines (say, from the NE, NW and SE points) to $\{0,1,\infty\}$, and you still have an arbitrary point (from the SW) left over.