I've always read there are two ways to look at a dim chord.

One is as an incomplete dom7. So a Bdim is basically a G7 without the root. But... I'm not seeing that in your examples.

The other is as a passing chord with no tonal center. In that context, each note of the diminished chord can be considered equally valid root. So a C#dim7 (C#,E,G,Bb) could be written with any of the notes as the root. What really matters is the note that the chord is leading to - optimally you want the notes to move smoothly to the target.

That makes sense for where it's used for a brief duration, but again, not for your examples, where it's held for a whole measure.

So... That sort of leaves that dim chord to be explained some sort of passing chromatic motion.

With the first two examples:

C | Cdim | Dm7 | G7
C | Ebdim | Dm7 | G7

if you treat the Cdim6 as a Ebdim6, they're the same chord - a double chromatic suspension resolving to the Dm7:

C -> C
Eb -> D
Gb -> F
A -> A

With the last progression,

F | F#dim | C/G | A7

I see this as more a passing bass note, along the lines of:

Fmaj7 (Am/F) | F#dim (Am/F#) | C/G | A7

So I'm not sure they're "related" to the key of C, in the sense that they're diatonic. But I'm a dilettante, not a practitioner.

-- David Cuny
My virtual singer development blog

Vocal control, you say. Never heard of it. Is that some kind of ProTools thing?