I'd really love it if you could give some intuition on how to derive the $x$, $y$ & $z$ coordinates from all/any of the orthogonal coordinate systems in this list, how you think about, say, bipolar coordinates if you had to re-derive the coordinate system on a desert island etc...
Apart from spherical & cylindrical I have no idea how to remember the others, these two I remember because I can think of the picture & re-derive how to express $x$, $y$ & $z$ in terms of $r$, $θ$ & $z$ etc... but the others are completely crazy geometrically & I have no intuition on them, & I need to at least learn how to represent $x$, $y$ & $z$ in terms of each system (so I can get grad, div, curl etc...) along with intuition on when to use them.
One interesting example of what I'm hoping for is with toroidal coordinates whose wiki is incomprehensible yet apparently there is an insanely simply way (page 114, also in this link) to derive a weak version of this coordinate system via a picture, any intuition on the rest of them?
The most important example, however, is confocal ellipsoidal coordinates (the easiest mathematical derivation of which, is given here). This aspect of the question has also been asked in this post, and it sets the tone for this question. The closest thing to a geometric interpretation for this system is given in Hobson, again though, no picture, merely a special case of what I asked for in my post and the pictures in my post as also special cases of the general case. Mathematically ellipsoidal coordinates are the most important because one can derive all the other coordinates from these by simple substitutions (a link to these substitutions in Morse and Feshbach is in the "in this post" link), ideally the goal will be to give geometric interpretations for all these substitutions as well (so we'd have two ways of getting all the coordinate systems!).
So: a) Draw a picture, b) Draw the projections determining the coordinates, analogous to this: