@aksalver: Lucas here with Rachio. Glad to hear the install went smoothly, though it sounds like you're not getting water when trying to run?
Blue status light, you say? That's a good sign. Do you know if your system had a Master Valve? From the pictures you provided (thanks for those!), it would appear you do. One thing you might be missing is a toggle on the app. You'll want to make sure your Master Valve is enabled in the App. (under Device Settings >> Advanced >> Master Valve/Pump)
Your wiring appears sound, but I do have a few recommendations. (below)
As others have noted, we do recommend one wire per 'C' slot . It shouldn't matter which goes where, though swapping them is always an option if you continue to experience difficulty.
(Often we see just one common unless there are sensors at play, but it really depends on your setup.)
Lastly, I notice a few wires that have the same (or similar) colors. I'm assuming you were careful in keeping these separate, but ensuring the 'P' (black) wire wasn't mixed up with one of the grey zone wires would be a final step if none of the above works.
(also, the blue Zone 4 is similar to what I assume was a power wire -- far left, 24VAC? If that were crossed, though - you'd likely be losing power or seeing much stranger behavior)
I hope that helps you get things up & running! Please let us know if you have any other questions or feedback.