What is the canonical name for the category whose objects are all pairs $(X, \rho)$, where $X$ is a set and $\rho$ is a binary relation on $X$, and whose objects are relation-preserving maps? That is, a morphism $(X, \rho) \overset{f}{\to} (Y, \sigma)$ is a map $f : X \to Y$ that satisfies $x \mathbin{\rho} x' \implies f(x) \mathbin{\sigma} f(x')$.
Abstract and Concrete Categories calls this category $\mathbf{Rel}$ (in fact the above definition was lifted from Page 22), but Wikipedia defines $\mathbf{Rel}$ as the category whose objects are sets and whose morphisms are relations between them. Why is there such a conflict of definition, and which definition is accepted?