I read the "Eb/C - D/C - Db/C" like, "bass stays at C and piano or guitar or whatever can play these chromatically descending major chords, in order to produce essentially the kind of barebones accompaniment that the composer or arranger had in mind."
Eb/C creates a Cm7, but spelling out the Eb, D and Db is a helpful hint for seeing how you can or perhaps even should look at it. It might also make some players play the chords with a bit more open voicings, not cramming all four notes of the Cm7 together inside one octave, but dropping the C to bass.
You could also take advantage of the chords written as Eb - D - Db by outlining or arpeggiating those triads in your solo? Why not even dissect the Cmaj9 as something like Em/C or Em7/C or G6/C? More possibilities for outlining chords as arpeggios.
Chord symbols should be seen as guidelines for accompaniment, not as theoretic harmony specifications for feeding into a musician-guidance autopilot computer that calculates the "correct" scales to use. :)
You can read and write chord symbols in many different ways, like:
- a pianist can play the thing before the slash with her right hand, and the thing after the slash with her left hand
- guitarist plays the thing before the slash, bassist plays the thing after the slash
- that's how the composer or arranger thought of it, separating a bass movement (or pedal tone) from stuff happening higher up
As a player and improviser you're supposed to understand the bigger picture, i.e. what it all sounds like when put together as accompaniment to the melody. Make up your own mind as to where the harmony could lean on at each point, and where it seems like going to. I kind of like the attitude that you shouldn't try to spell everything out and spoon-feed particular exact harmonies to jazz players who are supposed to be able to improvise their own arrangements on the fly. How often is it actually necessary to write a maj9, and a maj7 wouldn't be enough to deliver the essential idea, letting the players add 9ths and whatever seasoning they want? Isn't that like listing each individual note where a guitarist should use vibrato, and if he should pick with an upstroke or downstroke? :)