Let's consider a complex manifold $X$ of (complex) dimension $n$, and
equip it with some Hermitian metric $h$. It's not hard to see that there
is a unique connection $\nabla_{Ch}$ that's compatible with the metric
and whose $(0,1)$-part is the $\bar\partial$ operator. This is the Chern
connection of $h$.
Now let's view the manifold $X$ as a real manifold $M$ of (real)
dimension $2n$. The Hermitian metric is now just a
Riemannian one, and thus comes with a unique connection $\nabla_{LC}$ that's
torsion-free and compatible with the metric. This is the Levi-Civita connection.
A fairly natural question is when those two connections are the same,
that is, when can we expect the complex differential geometry of the
manifold to coincide with its Riemannian geometry? After all, we know a
lot about Riemannian geometry and it would be nice to exploit that
knowledge to study the complex geometry. This happens when the Chern
connection has no torsion, as one can work out, and we call the
Hermitian metrics that satisfy this condition Kähler metrics.
The link between this definition of a Kähler metric and the exterior
form one is as follows: Let $\omega = - \operatorname{Im} h$ be the
Kähler form of $h$, and let $\tau$ be the torsion tensor of the Chern
connection of $h$. If we take three holomorphic tangent fields $\xi,
\nu, \eta$, then
$$
\partial\omega(\xi, \nu, \overline \eta)
= h(\tau(\xi, \nu), \overline \eta).
$$
Thus $\tau = 0$ if and only if $\partial \omega = 0$, which is
equivalent to $d \omega = 0$ as the form $\omega$ is real.
I like this definition because it's quite natural from a
differential-geometric point of view. It can also motivate Hodge
theory, because on the Riemannian side we have the Hodge isomorphism
between cohomology groups and harmonic forms on compact manifolds.
If we ask what that isomorphism looks like on the complex side on a
compact Kähler manifold, we'll eventually invent Hodge theory.