The general definition for a wave equation is
$\dfrac{\partial^2 u}{\partial^2 x}-\dfrac{\partial^2 u}{\partial^2 t}=0$ (or $=f$)
for some $u \in C^{2}([0, \infty) \times \mathbb{R}^n, \mathbb{R}^N)$.
Now, I also read the definition that a linear wave equation is of the form:
$g^{\mu \nu} \partial_{\mu} \partial_{\nu} u+ a^{\mu} \partial_{\mu}u +bu =0$
where $g^{\mu \nu}$ are components of a smooth $(n+1) \times (n+1)$ matrix valued function, $g^{00} < 0$, $g^{ij}$ positive definite, $a^{\mu},b$ smooth $N \times N$ matrix valued functions.
Well, it's not unusual, that there are different definitions for something also also it's clear, that the second definition is a generalization of the first. Still, I'm wondering how common the second definition for a wave map is, since I didn't see it anywhere except some specific sources.
When thinking about a wave on any non trivial geometry the second definition would apply. A vibrating circular membrane is one physical example where the second definition applies. In that case it has the form
$$\left(-\partial_t^2 + \partial_r^2+\frac{1}{r^2}\partial_\theta^2+\frac{1}{r}\partial_r\right)u(t,r,\theta)=0$$
Where $u$ represents the lateral elevation. In this case $[g^{\mu\nu}]=\text{Diag}(-1,1,\frac{1}{r^2})$and $[a^\mu]=(0,\frac{1}{r},0)$. In general, on a pseudo-Riemannian manifold with a metric of signature $(-1,1,...,1)$ a Levi-Civita connection one can write a "wave equation" as
\begin{align} g^{\mu\nu}\nabla_u\nabla_\nu u &=-k^2u\\ g^{\mu\nu}\partial_{\mu}\partial_\nu u - \Gamma^\mu\partial_\mu u &=-k^2u \end{align}
Where $\Gamma^\mu=g^{\lambda\nu}{\Gamma^\mu}_{\lambda\nu}$ and the gammas are the Christoffel symbols.