"The root of the diminished chord is taken from the "generator" or the dominant seventh flat nine chord which is a major third below that root"--
When you are in the key of C and scrunch up a half step, you go to a C#diminished--yes, you are actually moving though the circle of fifths to the A dominant call it what you want (and into the key of D)--but you wouldn't want to name it that way on a chart is that you want the player to know that you are moving up from the tonic by half steps and not down by a minor third (which is what you will get if you punch an A7b9 or onto the chart--
On a score, which has all the part written out, the chord naming(which is, because all the parts are written out, academic) would follow the movement through the circle of keys, but on a chart, the chord names are there to give a sense of how the bass line is supposed to move--