I have to admit to being a bit lost as to what grades of abrasive exist in what media
Being a bit unsure of what equates to what is commonplace and even outright confusion is understandable because you can't trust comparative grit guides. Even doing a cursory investigation into abrasive stone comparison charts you should find some inconsistencies.
You might suspect this is an example of "You can't trust stuff on the Internet" but unfortunately it's more pervasive than that as this is also true of guides published in print, including those from the manufacturers themselves. So one has to be very careful about accepting any comparative information as definitive, although the defined grit size for any given product is more trustworthy (but not entirely reliable either).
Here are two charts giving comparative grit sizes:
[Source: Norton catalogue]
[Source: truncated from a chart found on Bladeforums]
As you can see they agree and disagree about equally. Worth also comparing these to the chart linked to in rob's answer.
and how either compares to any of the natural stones which are referred to only by name..
It's worth noting that natural sharpening media, just like wood, are inherently variable. Any given grit rating for a type of stone like Soft Arkansas or Belgian Coticule should be taken as an approximation at best.
In addition to some difference in apparent grit size they can vary in how well they cut, due to differences such as the grit:matrix ratio as well as in the variation of grit sizes within the stone (there is nearly always a mixture, with some or many smaller particles amongst the coarsest ones). Books going back to the 19th century mention the variability, e.g. that you might find "a quick-cutting [example]" of a finer stone.
"Fine" and "extra fine" seem to be defined differently for every manufacturer or medium, and not all of them give numeric values on the abrasive scale which might make them comparable...
Very true. Even for the same type of sharpening medium two manufacturers might have a slightly different grit size in products of the same name. FWIW and just from casual observation this seems to be spread fairly evenly from the fine to the coarse end of the market.
Is it really just "go by the numbers and stop when you're satisfied with the edge"?
To a degree, yes. But this is a little misleading as the fact is most people over-sharpen some or all of the time.
It's best to adopt a KISS approach to sharpening and not over-think it (very common in the West). First, what is the goal? To grind two faces to meet at a sharp edge, ideally with a zero radius. The more refined that edge, the less irregularity present, the sharper it is.
If that is firmly borne in mind it's much simpler to sharpen appropriately. It's also good to separate the concept of sharpening from honing.
Sharpening is edge creation. Sharpening is for new tools that don't yet have a proper edge, and when an edge needs to be created from scratch (pardon the pun).
Honing is edge refinement. Honing is the last step in sharpening but it should be the only step required when your knife, chisel or plane irons have lost their keen edge. So honing is what you do between sharpenings, meaning that in fact most of 'sharpening' is actually honing (or should be), since you can hone dozens of times between sharpenings so it is done far far more frequently.
If you hone well, and with sufficient frequency, you might need to sharpen yearly or less.
When honing you shouldn't even glance at your coarser sharpening media, and in fact a medium stone would often be much more aggressive than the edge needs. Using one fine stone or diamond plate, followed by a strop, or a fine and ultra-fine stone if using waterstones, should normally be more than sufficient.
And in fact at the very simplest you can hone by stropping only. An edge that's not quite sharp enough can be brought back to shaving-sharp in just 20-30 seconds by stropping, this is commonly done with carving knives and should become more common with chisels and plane irons.