Given $e$ so that it satisfies your condition of distinct triangles intersecting nontrivially we will prove that it comes from an associated polygon by induction on the number of vertices of the polygon.
First pick an $i$ so that our internal point is not contained in the triangle $(i-1,i,i+1)$, where indices are mod $n$. Now from the induction hypothesis there is a polygon $P_1P_2\cdots\hat{P_i}\cdots P_n$ on $n-1$ vertices and a point inside it so that the data agrees with $e$ on these vertices.
Suppose $P_{i-2}P_{i-1}\cap P_{i+1}P_{i+2}=Q_i$, so that we have to choose $P_i$ inside the triangle $\triangle P_{i-1}Q_iP_{i+1}$. The only thing we need to take care of is that the line joining $P_i$ to the internal point intersects the appropriate edge of the polygon as described by $e$. e$ (This is also equivalent to making sure that among the vertices which are paired up with the edge $P_{i-1}P_{i+1}$ the appropriate subsets are paired with $P_{i-1}P_{i}$ or $P_{i}P_{i+1}$ after the point $P_i$ is added). Notice that the intersection condition implies that the edge prescribed to $P_i$ lies between those prescribed to $P_{i-1}$ and $P_{i+1}$. So it follows from a continuity argument that some point inside $\triangle P_{i-1}Q_iP_{i+1}$ satisfies the required condition.
