I was puzzling over the same question. Some of these words are semantically overloaded, and as far as I can tell there's no single interpretation that makes it all "work out" the way you would require if this were math. There are two meanings to "adding 7":

1. adding 7 arithmetically to obtain the same note an octave higher;

2. adding in a 7th note to whatever chord you are playing.

But the way the chords are named isn't what you'd guess from those two meanings. If it were, you'd add a 6th to C major to get C6 (that part is true) and then a 7th to C6 to get C13 (that part is not true). Instead, there's yet a third way the notes are "added", namely cumulatively in the following sequence (I've cribbed most of this from the OP):

  C      C+E+G
  C7     C+E+G+B♭ 
  C9     C+E+G+B♭+D 
  C11    C+E+G+B♭+D+F 
  C13    C+E+G+B♭+D+F+A

(The author's argument is that while the chords are theoretically defined to include all these notes equally, some notes - especially B♭ - are more important than others. So even if you drop E,G,D,F and only play C+B♭+A, that's still arguably C13. He's saying that's the essence of the chord.)

I suppose some arbitrariness is inevitable, because there are only so many letters and numbers to go around, and more chords with a plausible claim to the "best" names -- which, if you know regular expressions, would be something like [A-G][1-9]+ -- than there are names available. One beautiful thing about music is that it isn't math (we have math for that). It seems to overlap with math and then mixes everything up in mind-blowing ways, at which point all you can do is feel. Well, memorize and feel.

Does anyone know historically how C13 got to mean "C+E+G+B♭+D+F+A" rather than "C+E+G+B♭+A", which if you consider things only symbolically, seems more likely? Obviously, it must be because that's what people were playing - but who? Presumably jazz musicians? When was this name settled upon?