Per the title, I'm seeking the definition of a function $f(n, m)$ which evaluates to the number of lines made from exactly $n$ points which can be placed on a two-dimensional discrete, square grid of size $m \times m$.

I've opened a separate question for the case where $n = 3$, as this is the answer I'm really after. I would've raised this question alone, however since I see that $n = 2$ has been answered already, I figured I might as well just generalize the question in hopes that there's some nice, elegant solution out there.