Choosing between semolina and non-semolina durum flour

Your understanding is mostly correct. Durum flour is sometimes used for bread, generally mixed with softer flour. (Pasta is also sometimes made from a mix of hard and soft flours.) It's unusual (in my personal experience) to find soft wheat semolina labeled as "semolina". More commonly it's labeled as "rava", "farina", or "cream of wheat". Labelling will vary by region.

Pasta can be made from any of the flours you listed. Using semolina will give you a firm but tender pasta; coarse semolina will give you a rougher texture (but still firm/tender). Using part or all bread flour will result in a chewier, stretchier pasta (think hand-pulled Chinese noodles). Pasta dough with coarse semolina should be rested for a while before rolling out, as it takes longer to hydrate.

Since we get variations of this question so often, I made a picture to clear it up at a glance.

Your understanding is pretty much correct. I would only add that the special "way" of converting wheat to powder is simply leaving coarser particles (for semolina) and smaller particles (for "standard" flour).

The question of "when to use which" doesn't have a well-defined answer. If you are experimenting, then any combination is good, see Sneftel's answer as to which product promotes which texture. If you have a recipe, then you should use the exact flour (or semolina, or mixture) that is defined in the recipe, else you won't get the result intended by the recipe.

The language part of your question is complicated. If I try to find exact words for each of the four categories, then I end up using "flour" for only the fine particle size, and needing the descriptive phrase "soft winter wheat" for the wheat which is used in making AP flour. This is not normal AE or BE usage though. People there tend to use "flour" (or "AP flour") for fine powder made from soft winter wheat, "semolina" for coarse powder made from durum wheat, and to never in their life encounter the other two products.

I cannot answer the part about buying the right product in Italy, somebody else would have to supply that part. You need some product which is made from "grano duro" if you want durum, but I can't say how to find the right particle size. Note that it is entirely possible that Italian language is also imprecise about milling sizes, and if that's the case, then you cannot get the right product just by the name.