I was reading about discrete metric spaces and got a bit confused as to what exactly is the correct definition.
The first one that I came across is that a metric space $(D, \rho)$ is discrete if the accompanying metric is discrete, meaning $\rho(x,y) = 1$ if $x \neq y$ or $\rho(x,y)=0$ if $x=y$. If I understand that correctly then also space of continuous functions $C[0,1]$ equipped with such metric is discrete?
The second definition stated that a metric space $(D, \rho)$ is discrete if there exists some $\kappa > 0$ such that $\rho(x,y)\geq \kappa$ for every $x, y \in D$. By this definition then also $(\mathbb{N}, |\cdot|)$ is a discrete metric space even though the metric here is the absolute value of difference.
Thanks for any clarifications in advance!
You are confusing the discrete metric with a discrete metric space.
Discrete metric
The discrete metric $\rho:X\times X\rightarrow [0,\infty)$ over a set $X$ is defined: $$\rho=\left\{\begin{matrix}0,&\text{ if x = y}\\1,&\text{ otherwise}\end{matrix}\right.$$
Discrete space
Let $(Y,d)$ be a metric space. $(Y,d)$ is called a discrete space if each $x\in Y$ is an isolated point. In other words, there exists a $\delta>0$ such that for every $y\in Y$ distinct from $x$ we have $d(x,y)>\delta$.
It is also clear that the discrete metric on a nonempty set constitutes a discrete space. I hope that clears up your confusion.