It's normal for the weak-link to fail first. This could be grip, arms, or probably a host of other muscle groups. This isn't a problem, it's more of an "opportunity" - at least you know which part to focus on to make improvements.
I didn't read the whole thread, so someone else may have answered. But I'll have a crack anyway.
Pull-up - palms facing away
chin-up - palms facing you ("supinated" grip)
semi-supinated - you grip handles that are perpendicular to your face as opposed to being parallel to your face. Also called a neutral grip.
They all work the entire upper back, shoulder muscles and arms. Some use biceps more or less than others and change the emphasis of shoulder muscles to some degree. I personally can't tell you which does which. I do know pull-ups are hardest, and chin-ups / neutral-grip pull-ups are easier. Most would probably say do all three if you can, and vary the width of grip too - justkeep mixing it up.
Well you lost me there for a bit, sorry. I think your definitions are different from mine. To quote Dire Straits: "when two men say they're Jesus, one of them must be wrong!". I don't know who's wrong, but I'll keep with my understanding of supinated/semi-supinated etc. for now until something better comes along.
I believe width of grip is not related to supination. Generally speaking, the wider the grip, the greater the workload is shifted away from your arms to your shoulders and/or back. Many will say that wide grip can cause shoulder joints to be stressed to the point where problems may occur (I am one of the many). I use a narrow-to-1.5 x shoulder-width grip most of the time, especially when I do weighted pull-ups. If I do wide grip pull-ups (1.5 - 2 x shoulder-width), I usually use body weight only to keep the risk of injury down.