Very interesting diagram. I hadn't ever seen the use of the "C-clef" to mark the pitch C in a space, rather than on a line. However, as @Old John points to in his comment, the Wikipedia page about clefs deals with just such a clef. At the top of the page this clef is first hinted at:
Only one clef that references a note in a space rather than on a line has ever been used.
And then later, this clef is revealed as being the C clef (most commonly used for the tenor and alto clefs, but also soprano, mezzo-soprano and baritone). However, in this case it indicates middle-C as being in the third space (from the bottom) of the stave. This, of course, means that all notes are identical in pitch to those when using the treble (G) clef, except that they sound an octave lower. It is not surprising that this clef is not commonly used to denote treble clef notes sounding an octave lower, as there are other, well established alternatives. Therefore, these two octave-transposing treble clefs, and the following "C" clef have the same function:
The first of these is commonly used, for instance for a tenor voice part or for an octave transposing instrument such as guitar (although a normal treble clef is also used, as the transposition is implicit). The second clef is less common. And, as I say, I had not seen the third clef before.
The Wikipedia page points out another good reason not to use this clef:
the C clef on the third space, [is] easily confused with the tenor clef
However, this doesn't explain a couple of aspects of the diagram you show.
- It isn't clear if it is necessary to place a standard treble clef before this "C" clef. I suspect not, and that the treble clef is written there to show that they denote equivalent pitches, although at different octaves.
- It isn't clear at which pitch the C between the staves should be: as @Brian Chandler points out, there is no indication that the lower bass (F) clef is an 8vb (octave lower) clef. For this reason, the C between the staves is in fact two different notes at the same time! (It is middle-C if relating it to the bass clef; it is an octave below middle-C if relating it to the clef on the upper stave.)
Although I'm sure the book you found this example in is a great resource, this example is certainly ambiguous, for the reasons stated above. (Having read the rest of @Brian Chandler's answer, I wonder if the musical example you show is in fact different to that found in the book?)