Welcome to the fold, Milo!
Part of the reason some of the terms may seem like an alien language is that some of them can actually change meaning depending on the arena in which you use them.
Case in point: "French" or "Triple" milling soaps.
When you see these terms in relation to, say, a handmade soapers forum, generally we're talking about rebatching, not melt and pour. Rebatching is a method sometimes used to "save" a batch of cold or hot process soap that failed due to any number of reasons. Another reason (and what I personally look to achieve) is to be able to utilize ingredients that may react unfavorably with the harsh, alkaline environment of fresh soap and thus assure that more of those ingredients touted properties end up in the finished product.
However, technically only a commercial soapmaker with the proper equipment can actually make French-milled/Triple-milled soaps. The milling process was originated by French soapmakers (hence French-milled). Using Stainless steel rolling machines, (think giant pasta roller/fondant sheeter) they would send freshly made soap through the rollers three times (hence Triple-milled). This would work and compress the soap into a paste, which was then extruded and cut or pressed into molds to set up. The result was a high-quality, low-impurity, smooth soap product.
Hope this helps some!