Just confirming that the solenoids are 24 volt AC and not DC (although that threw a different error for another user recently).
So the two white wires were in what appears to be a wire nut in the Rainbird picture then?
I noticed that none of the blue lights are lit on the Rachio unit. Are all four lights solid when the Rachio is plugged in?
@Gene provided the link to how the wireless rain sensor should be configured on the Rachio unit.
I might try putting the white wire from the field cable in the C(ommon) terminal on the other end of the terminal strip by the 8 terminal to see if that makes a difference. Also, try running a zone without the white wire from the field cable in the C(ommon) terminal and nothing should come on.
I've seen the folks in the wiring queue at firstname.lastname@example.org solve lots of wiring issues, so if none of these suggestions work don't be afraid to drop them a line to ask for assistance (if you haven't already) as they have access to diagnostic information from the unit that Gene and I don't have access to.