Exact category in the sense of Barr

Remarks

The second part of the first clause says that $(r_1, r_2) \colon R \to X \times X$ is a monomorphism. The kernel pair of any morphism is always an internal equivalence relation, so clause 2 asks for the converse.

Example: the category of sets is Barr-exact. An equivalence relation $R \subseteq X \times X$ is the kernel pair of the quotient map $X \to X / R$.

Non-example: the category of torsion-free abelian groups. In $\Z$, the subgroup $R = \set{(a, b) : a \equiv b \pmod 2}$ of $\Z \times \Z$ is torsion-free and is an internal equivalence relation. Let $f \colon \Z \to Y$ be a homomorphism into a torsion-free group with $f(2) = 0$. Then $2 f(1) = 0$, so $f(1) = 0$ and the kernel pair of $f$ is all of $\Z \times \Z$. So $R$ is not effective and clause 2 fails.