Why do we define compactness the way we do?

That definition is the more general definition - it holds in $\mathbb{R}^n$ and in things very different from $\mathbb{R}^n$. (In fact, the more concrete definition isn't even appropriate to arbitrary metric spaces!)

Specifically:

• It makes sense in arbitrary topological spaces, even ones which are non-metrizable (that is, which cannot be thought of as coming from a metric). For example, we can say with confidence that the cofinite topology on an infinite set is compact ... even though such a topological space is never metrizable.

• Within the context of arbitrary metric spaces, "closed and bounded" doesn't behave the way it should: consider a discrete metric space where every point is at distance $1$ from every other point. Every set in such a space is closed and bounded, but we don't have any of the phenomena associated to compactness in $\mathbb{R}^n$ which we actually want (e.g. we can have an infinite sequence with no convergent subsequence). The open cover definition, by contrast, gets things right (e.g. a subset of a discrete metric space is compact iff it is finite).

The question "why we define something as something?" is a really tricky question. There is no some universal reason not to define "compact" as "closed and bounded" or as "finite" or as "empty" or as "green grass". It's just a definition, a label, nothing more.

What we actually care about is behaviour and usefulness. In the metric world it turns out that the property "every sequence has a convergent subsequence" is a very strong and desired one. We called that "(sequentially) compact". And by the Bolzano-Weierstrass theorem this definition is equivalent to being "closed and bounded" but only for $\mathbb{R}^n$. The simplest counterexample in the metric world is $\mathbb{Q}$. Indeed, not every sequence in $A=[0,1]\cap\mathbb{Q}$ has a convergent subsequence (e.g. approximation of $\sqrt{2}/2$ by rationals) even though $A$ is both closed and bounded (with respect to the Euclidean metric) in $\mathbb{Q}$.

So "every sequence has a convergent subsequence" is a great property. And in fact it is easily generalizable to non-metric spaces. In the general setup it is also known as "sequential compactness". But it turns out that for metric spaces there is another property that is equivalent to compactness, namely "every open cover has a finite subcover". And since the definition doesn't require a metric (unlike "bounded" property), it is easily generalizable to non-metric world as well. But unfortunately outside of the metric world, this definition of compactness is not equivalent to sequential compactness (in fact neither implies the other). Comparing the two definitions mathematicians came to conclusion that the "open cover" definition is actually more useful and hence it became the standard one.

It is a more intuitive definition

Well, just because something is more intuitive doesn't mean it is better. Besides, nothing is intuitive until it becomes intuitive. :) I doubt that any professional mathematician nowadays would call the open cover definition as counterintuitive. It is so common that they've got used to it.