I'm writing a chord identifier based on notes and need to know if it is possible or not.
I think this is the question really to be answered here...
I assume we're speaking about jazz/popular music chord names (as Dekkadeci pointed out, other naming conventions exist).
Chord symbols are not really meant to indicate the exact notes to be played by the musician. They leave room for interpretation: dropping or adding notes and choosing the actual voicing. A skilled musician may reharmonize the song on the fly. On the contrary, I guess you want the chord finder to output symbols as accurate and unambiguous as possible. It should be noted that these will likely not be the same as chord symbols typically used in practice.
It is very common for musicians to omit notes in chords. Even a simple Cm can be played as two notes, C and Eb. Especially in very small ensembles with one–two musicians, harmony movement might be barely hinted. C7 might easily become E-Bb. Chords might be played melodically rather than harmonically. Guessing the underlying harmony might be an impossible task for a program that only interprets the vertical structures.
Omitting notes is even more common in larger bands. A guitar may play Eb-G-Bb, which one would call Eb, but if bass plays C, it adds up to Cm7.
Chord naming convention (and music notation in general) is based on practice and historical conventions, rather than logic. It is also very much focused on western popular music. What it implies, it's not really feasible to label any given group of notes with a compact chord symbol. Chord symbols are meant for chords that are actually used in western popular music.
In particular, I can't recall seeing a minor chord with flat 9 in jazz standard repertoire. You'd think that it's a diatonic chord built on the third step of a major scale... but somehow it's not used. So I'm following more intuition than experience. I'd say that Cmb9 does contain flat 7. I'm not sure how would I play it as a musician. Normally fifth is the first one to go, but in this case it creates a very interesting dissonance with b9, so maybe I would drop Eb? Also, C-Eb-G-Bb-Db could be interpreted as Eb13... in such ambiguous cases the voicing may matter much for how the chord is interpreted.
Concerning Cmb13, a minor chord with 6 is typically written as Cm6. I guess you could write Cmb6, but again, this is not a typically used chord (at least not in pop/jazz repertoire)... and C-Eb-G-Ab might likely be heard as Abmaj7.
does it mean that the b13 is an addition and thus the 7th, 9th and 11th are NEVER played?
b6 is addition, but it won't stop a musician from adding 9, or b7 if they feel like it... I think Laurence summarized the naming/performance rules well.