Let $R$ be an integral domain and $S$ a subrng of $R$.
Definition 1.
$S$ is a subdomain of $R$ iff $S$ is an integral domain
Definition 2.
$S$ is a subdomain of $R$ iff $S$ is an integral domain and $1_R=1_S$.
Which is the proper definition in categorical view? Definition 2?
Moreover, are they equivalent?
These two definitions are equivalent.
Since $1_S\cdot 1_S =1_S\cdot 1_R$, $1_S(1_S-1_R)=0$.
Since $R$ is an integral domain, $1_S-1_R=0$.