Distributivity is somehow related to counting: $a\times 1 + a\times 1 = a\times (1+1) = a\times 2$. This excludes sets of non-numeric functions from forming a ring, since if $(S,\circ)$ is a group with $S$ a set of non-numeric functions, $f\in S$; let's denote by $\square$ what would be a "multiplication", $f\circ f\circ f= 3\square f$ but $\square$ is not a internal operator because $3\not\in S$. So it seems distributivity is related to counting, and that excludes many sets from forming a ring.
A ring could have been defined as a $(R,\clubsuit,\diamondsuit)$ such that $(R,\clubsuit)$ is a group and $(R,\diamondsuit)$ is a monoid, in which case $\clubsuit$ and $\diamondsuit$ could be independent and $R$ could be a set of functions.
So is relation between distributivity and counting correct, and why is it chosen? Note: rschwieb already gave a very reasonable reason.