Definition of uniformly cauchy In this post, the definition of uniformly Cauchy is defined as:
Uniformly cauchy sequences

A sequence of functions $f_n$ is said to be uniformly cauchy if
  $$\forall \varepsilon > 0 \ \exists N > 0 :\forall z , \forall r, s >
 N:  |f_r(z) - f_s(z)| < \varepsilon$$

On Wikipedia, uniformly Cauchy is defined as:
https://en.wikipedia.org/wiki/Uniformly_Cauchy_sequence

In mathematics, a sequence of functions $\{f_{n}\} $from a set S to a
  metric space $M$ is said to be uniformly Cauchy if:
For all $\varepsilon > 0$, there exists $N>0$ such that for all $x\in S:
 d(f_{n}(x), f_{m}(x)) < \varepsilon$ whenever $m, n > N$. 
Another way of
  saying this is that $d_u (f_{n}, f_{m}) \to 0$ as $m, n \to \infty$, where
  the uniform distance $d_u$ between two functions is defined by
$d_{u} (f, g) := \sup_{x \in S} d (f(x), g(x))$.

Question: When do we use absolute value $|\cdot|$ and when do we use uniform distance $d_u$. Are they equivalent?
 A: The first definition is (implicitly) for functions with image the real (or possibly complex) numbers, where the absolute value makes sense. 
In this case the most common way to define the distance of two values is by saying $d(s,t)= |s-t|$. 
More generally, if one has a normed spaced, like $\mathbb{R}^n$ with the euclidean norm for example, then a distance of two points $s,t$ can be defined via $d(s,t)= \| s - t  \|$. 
Thus the second definition, its first half, is more general than the first definition in that it works for any metric space, while the former needs an absolute value or at least a norm. Having an absolute value or a norm, one can define a distance, but not the other way round. And, if one does that in the usual way, the first definition becomes the second.
The first and the second part of the second definition are completely equivalent. (One could also write a version of the second part for the first definition.) What one uses is a matter of style and what one wishes to emphasize. 
A: If there was no universally agreed upon definition, you would use whatever metric is most suitable for a particular subject you're studying. For example, if you are working with $L^{1}(\Omega)$ space, you would use its corresponding metric. 
But I should point out that the uniform metric $d_{u}$ as it is defined in your question would give the same result, since the definitions fix the point $z$ or $x$ as $m,n$ vary. Thus, even with the uniform metric, you're only considering the distance between functions at a fixed point, then taking the supremum over all points. The first definition says this implicitly with the "for all $z$" part, then uniformly bounding the difference at each point by a uniform epsilon. Thus, they are the same definition. 
