...it seems to me that the main reason that most of those that like to list "out of the pot" ingredients is that they don't like the idea of listing sodium hydroxide...
This is not necessarily true. Some governing agencies require an "out of the pot" ingredients list -- I believe the EU is one.
Those in the US who don't want to list sodium hydroxide usually still want to list the starting fats, not the saponified results. So the "what comes out of the pot" method isn't any more appealing to these soap makers as the "what goes in the pot" method. Instead, they use all kinds of creative approaches to having their cake and eating it too. Like "saponified fats of ..." and variations on that theme.
In the US, there are no requirements for an ingredients list for true lye-based soap IF it is sold with no claim other than "it just gets you clean". You don't need any ingredients list at all if you don't want to use one. Or if you do, you can use that "saponified fats..." verbiage if that's what you want.
Once there are any cosmetics claims made for the soap, however, then a proper ingredients list is required. In the US, you can use a "what goes into the pot" method or a "what comes out of the pot" method, but you have to use one or the other.
If you make a drug claim, the rules are even more stringent.
Marie Gale and the US FDA website are the go-to sources of info on this topic.
