I've ran into the term "Higher Genus Hyperbolic Surface" in the context of Ricci flows of orientable surfaces of dimension $2$.
I'm not familiar with the term, and wasn't able to find a definition in Wikipedia or in any of the books on my desk.
I imagine the definition would be something like:
Connected sum of $g$ tori, endowed with a metric of sectional curvature $-1$.
Is this a correct definition of such a surface? Is there a nice way to visualize it?
Edit: More context.
I've ran into the term mentioned above while reading a discussion called "The Topology and Geometry of Low-Dimensional Manifolds" in P. Topping's "Lectures on the Ricci Flow".
The discussion aims to motivate the development of the Ricci Flow, as a tool for classifying such manifolds. It's informal in flavor. We assume all manifolds are compact and orientable. See bellow for summary.
The writer mentions that 2-dimensional manifolds are classified by the genus. After discussing that fact we're told:
It turns out each of such surface can be endowed with conformally equivalent metric of constant Gaussian curvature ... the universal cover of the surface must be $S^2,\mathbb{R}^2,\mathbb{H}^2$, and the original surface is than described as quotient of it's universal cover by a group of isometries acting freely.
This gives rise to $S^2$, a flat torus, or a higher genus hyperbolic surface, depending if the curvature is $1,0$ or $-1$ (up to uniform scaling of the metric).
From the quote you gave, it looks like "higher genus" simply means "genus higher than $1$", as suggested by @TheSilverDoe. Your description of such a surface is therefore appropriate if you add the requirement $g > 1$.
Learning to visualize such surfaces and their hyperbolic structures is a big, big topic. A good starting point is Jeff Week's book The Shape of Space. You can also pick up any textbook on hyperbolic manifolds.