"distributive property" vs. "ring homomorphism": comparing definitions

This is a good observation. The key here is that we want multiplication and our homomorphisms to be compatible with our addition operation. When we study groups, we have a set $G$ equipped with a single binary operation $G\times G\to G$ satisfying elementary properties: associativity, existence of inverses, and existence of an identity element. When we study rings $A$, we have two operations $+$ and $\times$, namely addition and multiplication. Usually $(A,+,\times)$ is required to be an Abelian group under $+$ and a (often commutative) monoid under $\times$ (inverses might not exist under multiplication, but the other properties of a group are respected).

The natural question remains: how should the operations interact with each other? If there is no interaction between the multiplicative and additive structures of the ring, we might as well separately study $(A,+)$ and $(A,\times)$ as an Abelian group and a monoid, respectively. So, we require that $\times$ "preserve" the group structure of $(A,+)$. A sensible interpretation of this is the following: the map $\phi: A\times A\to A$ given by $(x,y)\mapsto x\times y$ should be "like a homomorphism of $(A,+)$ to itself." With suitable modification, it is. If we fix $x$, then the map $$ \phi_x:(A,+)\to (A,+)$$ has $$ \phi_x(y+z)=\phi(x,y+z)=x(y+z)=xy+xz=\phi_x(y)+\phi_x(z)$$ and if we fix $y$, the map $$ \phi^y:(A,+)\to (A,+)$$ has $$ \phi^y(x+z)=\phi(x+z,y)=(x+z)y=xy+zy=\phi^y(x)+\phi^y(z).$$ So we have a pair of group homomorphisms associated with right and left multiplication.