any actual advice?
R.I.P. Les Paul, 1915-2009

A man chooses, a slave obeys.
That is a wine stain finish, then probably oil finished over the top.

Might be a matt or satin lacquer, but I would bet oil.

I'm not sure about this distinction between "Stain" and "Dye" that I keep seeing. AFAIK, there is no difference.

Can someone clarify?
Well, I have some experience staining, but not staining guitars. But a guitar is just a hunk of wood so there's really no difference. It looks like a deep red stain. You might be able to get that effect by purchasing and testing various stains. Just try it out on scrap pieces of the same material. Brush it on and then wipe it off with a rag right after to get rid of any brush marks or build-up. Make sure you wipe the rag in the same direction and at the same pace all the time. Wait a day or two and then repeat if you want it darker. It might be clear-coat, you could certainly do that, but it isn't shining so it might just be a matte finish.

Don't just take my advice, do some research on the internet if you're thinking of doing this yourself, or you could ask some of the luthiers in residence. You can never know enough.

As for the difference between a dye and a stain, a dye is a coloured substance that has a "chemical affinity to the substrate." What that means is that it likes to bond or adhere to the surface in which it is applied, this case being a guitar. Dyes are a main ingredient in stain, the other two being the solvent and the binder.

So, in summary, a dye is part of a stain.