Trying to prove:
There is finite set of triangles with vertices on a grid enclosing any finite set of grid points. While there are infinite set of $n$-gones enclosing the same set for $n\ge4$. Some samples follow: for 1 point: $n$-gons on grid with $1$ internal grid point
For 2 points the triangles are:
And may be few more. There is a good sample of infinitness of enclosing 4-gones in comments of referenced question.