Clive can tell us about using salt to stabilize a print instead of hypo to fix it ( which is very interesting and worth a whole thread by itself! ). I think Kevin is referring to recommendations of a salt bath before fixing in hypo.
The salted paper print starts with some excess silver nitrate. The idea of the salt wash is to convert any remaining silver nitrate to silver chloride, so that it can be removed in the hypo. If there is still silver nitrate on the print when it goes into the hypo, it can cause stains, and it will decompose the hypo sooner.
You are absolutely correct that some instructions put the salt wash first and some put it after a water wash. Not only in modern descriptions of salt printing but in historical descriptions as well. Some instructions leave out the wash in salted water completely.
I've tried it every way. When I first started, I just washed in water and then fixed. Later, I read Chris James' book and started using the salt bath as the first step. These days I usually "pre-wash" in water to remove most of the silver nitrate, then a brief 3 minute dip in salt water to convert what little silver nitrate remains to silver chloride, and then an extended water wash before fixing.
There's also some controversy or inconsistency about how strong the salt bath should be. I've been using a "pinch" of salt like James recommends, but many people use a much stronger solution ( 2% or even 10 or 15% ). I think most of these differences come from different ways of proceeding. If you don't pre-wash in water, there will be more AgNO3 to convert, so a weak salt solution might not be enough to convert it all, especially if it's sunk into thick paper. Also the amount of excess silver will depend on a lot of factors, like whether you double coat, what kind of binder, how thick the paper is, etc.
Most of the time, with thin papers at least, I have not seen any difference between these different approaches. A good water wash, or 1st in salt then water, or 1st in water and then in salt all can work well with no staining. Where it could matter is if you are having trouble. I don't want to write a whole book here, but depending on the cause of the stain, one washing approach might work better than another. But always the first thing to try is longer washing!