I don't think the order of adjustments is critical. These adjusters are usually interactive -- that is, changing one adjuster also affects another parameter somewhat -- so you often need to go back and forth between adjusters to get the best screen geometry.
For example, adjusting the horizontal width may affect the horizontal linearity, and vice versa.
Your image looks like it is not quite centered horizontally. You could start with the horizontal centering adjuster and try to move the whole image somewhat to the left. After it's centered, then you can play with horizontal width and linearity to get the best compromise. Decrease the width until you can just see black bars on the edges, then increase it just enough to fill the screen horizontally. That may change the linearity a little, so touch up the linearity and see how it looks. If the centering seems to have changed a little, touch that up again. And so on. As you go back and forth between adjusters, you'll learn how they affect each other.
I use the term "compromise" because on some TVs the screen geometry will never be absolutely perfect. The horizontal non-linearity in your picture is noticeable in a test pattern, but you probably wouldn't mind it as much when watching actual programs. If you get frustrated with adjustments, take a break and watch an old movie for a while. That will give you a better idea how much a particular geometry imperfection affects your viewing enjoyment
Phil Nelson
Phil's Old Radios
http://antiqueradio.org/index.html