The Cdim7(b13) is a diminished chord with a diminished 7th (practically equals a major 6th) AND an additional minor 6th. Because the minor 6th is extending the dim7 chord rather than replacing the 7th, it is written as b13 in the chord symbol and not as b6.
A Cdim7(b13) has the notes C (1), Eb (m3), Gb (b5), Bbb (dim7) and Ab (b13). On the guitar the diminished 5th (b5) will often be abandoned to be able to play the b13.
Already mentioned: the diminished 7th Bbb practically equals a major 6th A, but especially when talking about chords like this one you have to be aware of the fact that you should never relate to the same note or interval more than once. That means 6 and b6 respectively A and Ab should be avoided in the same chord symbols, scales, etc. When talking about a plain dim7 chord, this conflict doesn't happen that much.
The strictly analytically „thinking“ chord analyzer can’t think „actually it would be a …, he just didn’t play the …“. Consequently it would take the dim7(b13) without the b5 for a m6(#5) chord, which is syntactically right, but falls short of the context.
A Cdim7(b13) can normally be played instead of a plain Cdim or Cdim7 just like that.