How to choose: which V-I motion do you wanto to imply? Depending on what your "V" chord is, you choose the corresponding substitute dim chord.
I use a simple way for looking at dim (or dim7) chords: one of the notes works as the third of some major - or major 7th i.e. dominant 7th chord. This logic comes as a consequence from the fact that you can substitute a dominant 7th chord with a dim7 that includes the third of the chord. For example, you substitute a G7 with Bdim7 (or Ddim7 or Fdim7 or G#dim7). There's no single correct answer - any interpretation works.
So, for your example, the C and then Cdim or C#dim question. Following the rule of thumb I presented above, as which dominant sevenths can a Cdim work? :
- (1) Ab7 (imaginary tonic being Db or Dbm)
- (2) B7 (imaginary tonic E or Em)
- (3) D7 (imaginary tonic G or Gm)
- (4) F7 (imaginary tonic Bb or Bbm)
In the context of the key of C major, interpretations 2 and 3 feel logical to me. For example if the chords are like, C - Cdim - Dm, you could treat the Cdim as a B7 secondary dominant intending to push to E7, before the harmony changes its mind and takes Dm instead. As melody or "solo", try playing whatever you would play on top of a B7, on the Cdim.
Then in the same way, let's see as which dominant sevenths can a C#dim work? :
- (1) A7 (imaginary tonic being D or Dm)
- (2) C7 (imaginary tonic F or Fm)
- (3) Eb7 (imaginary tonic Ab or Abm)
- (4) F#7 (imaginary tonic B or Bm)
In the context of the key of C major, interpretation 1 is the most natural for me. I see the chord progression C - C#dim - Dm as almost the same as C - A7/C# - Dm.
I think this is a very simple and practical way to deal with dim chords, interpreting them as being different-tasting dominant chords. Or offering ways to jump to different tonal interpretations. It's kind of a game of illusions and tricks - where's the tonic now? Are you sure? Where is it now? :)