So, I decided to break this down into small steps and start with perfectly polished obsidian.
I also decided to ask Blender for help and created a polished obsidian material. At first, I was struggling to understand what I was trying to create, but the deeper I looked, the more I realized just how reflective polished obsidian actually is. In fact, the "apache tears" on Geology.com are presented on a white surface with a quarter next to them, and the quarter is almost perfectly reflected. It's also reflected "under the surface" - subsurface scattering! Polished obsidian pictures on Google (none of which I want to link because they are either from retail stores or wildly inappropriate) all seemed to agree - it's a black, glass-like material that confers near perfect, sub-surface reflections.
I'm looking at my black, shiny plastic monitor stand and wondering right now what about it makes the visual difference evident. Hrm.
Anyway, here's a Blender render of my material. The diffuse bsdf is 100% rough and the gloss (Ashikhmin-Shirley) has .01 roughness and a 100% white color. The mix shader is set with an unmodified Frasnel input. The diffuse is mixed with a subsurface scattering shader (Christensen-Burley) with a scale of .5, and other values default. The mix here is likewise controlled with the Fresnel because I reasoned that the subsurface scatter is probably less evident at near horizontal angles from the viewer. The HDRI is of De Balie Theatre, where the Blender conference is held and was provided freely by Thomas Radeke, one of the 2016 attendees (though I no longer seem to have a link. I'll have to rewatch the open-mic session to find it. :/)
Blender Cycles Render
Even if this isn't accurate, I feel like it's helped me make better sense of the reference images I've been studying.
My next step will be to attempt to paint polished obsidian, though I'll place it in a simpler setting. Then, I'll work on adding streaks of brown color. Finally, I'll work on making it look chiseled.