I read on a paper that a relation $\mathcal{R} \subset \mathbb{R}^n \times \mathbb{R}^n$ is monotone if
\begin{equation} (x_1 - x_2)^\top (y_1 - y_2) \geq 0 \end{equation}
holds for any pair $(x_1, y_1)$, $(x_2,y_2) \in \mathcal{R}$. This is also found on Definition 20.1 in the book "Convex Analysis and Monotone Operator Theory in Hilbert Spaces" by Bauschke and Combettes, 2017.
In other papers (e.g. this, Definition 3.6 ), I saw monoton relations in $\mathbb{R} \times \mathbb{R}$ defined saying that the relation is monotone if $x_1 \geq x_2$ implies that $y_1\geq y_2$?
I do not understand the connection. The second definition is more familiar to me, since it is similar to monotone functions (I have an engineering background). Is there an order relation for vectors that leads to the inner product of the first definition?