It would be good to keep the concepts of modality and chord identification separate. First, Tim's answer above explains that it is common to add other notes to chords for color, and thus the chord is indeed C9 the whole time. See his answer for more explanation.
As for the idea of this being a modal progression, that's sort of another thing altogether. You would need to see more of the melody, especially the cadence, to determine whether or not it is in fact modal. If that Bb is persistent through the entire passage, and if C is indeed the home tone, then this is also a modal composition. If The Bb is not constant, or if C is not where the melody comes to rest at the end, it is probably not a modal composition.
But I would keep the two worlds separate: (1) What is the chord? This is good to know if you are analyzing how the chords work together. (2) What mode is it in?. The requires analyzing the melody and determining what it's tonic note is, and then considering the other notes around it and whether the resulting mode is major, minor, Dorian, etc.
To take it a step further, there are specific things you need to do to write modal music. Mostly you are attempting to keep your composition from straying back into major/minor territory. But that is a separate topic.