Any ideas on what made
$$\forall \varepsilon>0,\exists n_0 \in \mathbb{N}\text{ such that }m,n \geq n_0 \implies |x_m-x_n|<\varepsilon$$
prevail over the equivalent and arguably simpler (for making use of one less variable) $$ \forall \varepsilon>0,\exists n_0 \in \mathbb{N}\text{ such that }n \geq n_0 \implies |x_n-x_{n_0}|<\varepsilon $$ as the standard definition of a Cauchy sequence?