From Wikipedia, a contraction mapping is a function $f: M \rightarrow M$ on a metric space $(M,d)$ such that there exists a nonnegative real number $k<1$ such that for all $x,y\in M$, $$ d \left(f(x),f(y)\right) \leq k \cdot d(x,y). \tag{1} $$
I have a question that's probably pretty basic since I feel that I just have the wrong intuition for it right now.
Why could the above definition not just be replaced (unless it can be) with a simpler definition like this?:
A function $f: M \rightarrow M$ on a metric space $(M,d)$ is a contraction mapping if and only if $$ d \left(f(x),f(y)\right) < d(x,y) \tag{2} $$ for all $x,y\in M$ (replacing the weak inequality and constant $k$ with just a strict inequality)? This is probably just an intuitive misunderstanding, but to me, it seems like that a contraction map is just a function where given two points, they always get mapped by $f$ to two points that are even closer together.
In order to answer "why is this definition the way it is?" it helps to ask a more basic question: "what is this definition good for?"
The property $d(f(x),f(y))\le kd(x,y)$ (for all $x,y $) implies the existence of a unique fixed point of $f$, provided that $f:X\to X$ is a map on a complete metric space $X$. (As Daniel Fischer said in a comment.) In other words, the equation $f(x)=x$ has a unique solution. This is fantastic, getting the existence of a solution of an equation (where $x$ may well be a vector or a function) from a mild assumption. This theorem is currently high on the list of overpowered mathematics results, whatever that means.
The property $d(f(x),f(y))<d(x,y)$ (for all $x\ne y$) implies the uniqueness of a fixed point, should it exist. It does not imply existence, and uniqueness with no clue to existence is not nearly as useful. For a concrete example, $$f(x)=x-\arctan x$$ is a map from $\mathbb R$ to $\mathbb R$ which satisfies this property, but has no fixed points.