I think the reality is closer to different dysbiotic states can cause depression. Depending on the the specific dysbiosis, the interventions will be different.
.. And would you categorise those different dysbiotic states into ... types?
I'm sure you could - but if you went about it scientifically there would easily be thousands (not just the 3 in Ayurveda). That being said, I do think Ayurveda is mostly about manipulating the microbiome, and I think there are probably real nuggets of truth in its approaches. For example the use of Triphala to improve motility will help in numerous dysbiotic states.