Q:

What's the difference to a player between dim7 and 7b9?

I'm trying to figure out how best to write a lead sheet for a player given a certain chord progression as an example:
shortchord.org/RqxRx (disregard the automatic names this app gives to the chords)
It seems like I have two options for how to write chord symbols for this progression, with the difference being in how to represent the second chord.
I can see naming this progression in terms of chord function, where the second chord is a secondary dominant of Dm7, as:
Cmaj7 | A7b9/C# | Dm7 | G7
( I | V of ii | ii | V )
Or, I can see naming the chord to be more accurate about which notes this specific voicing contains:
Cmaj7 | C#dim7 | Dm7 | G7
Both describe the rising chromatic bass note from C to C#, but one describes the dominant relationship, while the other specifies the dim7 chord structure without adding the implied A.
My question is, which is easier for a player to understand? Or, at least, what is the difference between what's communicated to a player in each approach?

A:

The difference is in function.  The A7b9 is a dominant chord leading to Dm. The C# dim would imply a passing chord between Cmaj and Dm.
From the standpoint of improvisation, if the chord is really really the V of Dm then I'd prefer your first rendering:
Cmaj7 | A7b9/C# | Dm7 | G7
This would imply a particular scale - perhaps a mixolydian scale with a flatted ninth or the fifth mode of (d) harmonic minor.  As a player, I can still choose  to play a (C#) diminished arpeggio idea or a diminished scale here if that's what I hear.
However, if I see this:
Cmaj7 | C#dim7 | Dm7 | G7
I'm more likely to think C# diminished.
If the diminished chord is used as a passing chord, as in:
|Cmaj7 C#dim7 | Dm7 G7 |
(two chords per measure)  then I think the diminshed notation works better.

