I'm working through Ted Sider's book "Logic for Philosophy," and I'm noticing some discrepancies between the definition of a "Total Relation" that he uses and the definition used in other places, particularly Wikipedia.
The Definition from Sider's book: A relation $\mathscr{R}$ is total (in a set $\mathscr{D}$) iff for any $u,v\in \mathscr{D}$, $\langle u,v \rangle \in \mathscr{R}$.
The definition from wikipedia (translated into sider's notation): A relation $\mathscr{R}$ is total (in a set $\mathscr{D}$) iff for any $u,v\in \mathscr{D}$, either $\langle u,v \rangle \in \mathscr{R}$ or $\langle v,u \rangle \in \mathscr{R}$.
My question: which of these definitions is more standard? Or are both definitions used in different contexts? Perhaps, Sider's definition is common in metalogic and the alternative definition is common in mathematics.
Another note: Sider's definition is not the same as the alternative definition since the relation "$\leq$" would qualify as total over $\mathbb{R}$ by the alternative definition, but would not qualify by Sider's definition.
The second definition of "total" is more standard, as far as I know, both in mathematics and logic. Often times, we care about such total relations in the context of orderings: if our relation is a partial ordering (i.e. a reflexive, anti-symmetric, transitive relation), then saying the relation is total means it's also a linear order (and thus behaves like "$\leq$" behaves on numbers).
Sider's definition as you give it might be a called a "trivial" or "universal" relation, since it contains every ordered pair over the domain. This is sometimes an important extreme case to consider in proofs (along with the empty relation, or the identity relation).