It works for any polygon with vertices on the integer lattice, but what if the polygon doesn't have all vertices there? I was thinking, it might be possible to generalize it by scaling the lattice, so for example if a polygon has coordinates $(1,1), (1.5, 1), (1, 1.5), (1.5, 1.5)$ we could sort of shrink down the lattice, such that $1$ is now $0.5$ and then the new coordinates of the polygon would be $(2,2), (3, 2), (2, 3), (3, 3)$ now it has vertices in the integer lattice and we can now use Pick's Theorem like normal, and then scale down the area by half. I said I was shrinking down the lattice but you can also think of it as scaling up the polygon, multiplying all of its coordinates by a constant factor, such that all coordinates become integers and then using Pick's Theorem and then scaling down the resulting area by the factor we multiplied with.
That was at least my original idea, but this doesn't work, you can see it in the example I gave. The original polygon is a square and has area $0.25$, whereas the scaled polygon has area $1$, so 4 times the area even though we only scaled by 2. So my first question is, could this idea be fixed? If I have a polygon and scale it by a factor $\alpha$, what is the ratio of the area of the scaled polygon divided by the area of the original polygon in terms of $\alpha$? If I had to take a guess, based on the example I gave, it would be $\alpha^{2}$.
An answer to that question would fix the idea, however it would still not work for all polygons, since not all polygons can be scaled so that all their vertices are integers. To prove this, think of the x-distance between two points in the polygon (let's define x-distance for any two points $(p_{1}, p_{2}), (m_{1}, m{2})$ as $\lvert p_{1} - m_{1} \rvert$ and the y-distance as $\lvert p_{2} - m_{2} \rvert$, I don't know if these things have a name but that's what I am calling them).
Now consider the x-distance between any pair of points in the polygon, call it $\epsilon_{1}$, and the x-distance between any other pair of points in the polygon, call it $\epsilon_{2}$.Suppose it is possible to scale the polygon by $\alpha$ such that all vertices are in the integer lattice, this means our two x-distances which are now $\alpha\epsilon_{1}$ and $\alpha\epsilon_{2}$ are now integers, which means that $\frac{\alpha\epsilon_{1}}{\alpha\epsilon_{2}} = \frac{\epsilon_{1}}{\epsilon_{2}}$ is a rational number. This proves that if it is possible to scale a polygon into an integer lattice then the ratio between any two x-distances in that polygon must always be rational, but clearly an infinite number of polygons can be constructed such that is not the case, hence there are polygons that cannot be scaled into an integer lattice.
Is my argument about why scaling doesn't always work correct? Lastly just because it is not always possible to scale the polygon to apply Pick's Theorem directly doesn't mean that the theorem cannot be generalized to any polygon, does a more general version exist that works for all polygon with any coordinates?