Just some notes to your movie #2
The Fit to Object has no influence AFTER the Assign Coordinates to the UVW.
The Fit to Object without being in Texture Mode, will perhaps fit the size, but it doesn’t work for the orientation, that is not a given without your input.
You used the relax option again, which I would only suggest for the top and bottom, as the geometry is kind of folded: which is not nice in conjunction with the flat projection. If relaxed, you get some texture on all “polygons”.
As I wrote, for this object, perhaps avoid the UV manager completely, it is not really needed and as I can see, your work with it feels uncertain, trial and error. This needs more understanding, hence some tutorials or manual time.
Work with the move and scale (uniform or non-uniform) in UV mode, it is the best for now, and I use it even in more complex situations. Between 5:00 and 5:30 I can see that you get familiar with it, nice!
5:45 you ask about the little polygon, why is it not in the right position. Is my Lip-reading good enough? ;o) The reason is simple: the Cylindrical projection has a start and end point. You “can” is rotated exactly that the start an end is in the “middle” of the polygon stripe. Based in super tiny impressions (not yours!) it makes the decision to include a polygon at that start or at the end point. In your case on polygon was the start point.
You can select one single UV polygon and move it in position. To make that not trial and error, there is a snapping function Move Tool>Attribute Tab (UV Editor), this helps to move and snap the polygon into place.
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Please do this couple of times, check the steps that I have suggested and in which order they need to be applied. You should feel comfortable soon, and find maybe your own workflow then, as there are many ways to it.
If you feel comfortable with the result and the steps, get more info about the UV manager. It is a powerful instrument, and with that comes the danger to mess up things as well. But one step after the other, so you get comfortable with it, the more you know and have experienced the problems and options with these tools, the more fun you have to use them. With that, you will use them faster and more often.
My best wishes