How should Riemannian metric $g = e^xdx^2 + dy^2$ be interpreted?

Question

Suppose that $M$ is a Riemannian manifold $M$ in $\mathbb{R}^2$ and we are interested in what kind of metric does $g = e^xdx^2 + dy^2$ induce. Assuming that $g$ indeed is a Riemannian metric, should $g$'s precise definition be $g_p = e^{p_1}dx^2 + dy^2$ (where $p_k$ is the $k$th component of $p\in M$)? I have seen similarish objects as $g$ in couple of posts on this site and even in some textbooks, but I have trouble understanding how such a $g$ acts on the points, or point pairs, of our manifold $M$.