Step 1 - order an eight zone Rachio Gen 2 with an outdoor enclosure.
Step 2 - replace existing controller with the Rachio.
Sorry… couldn’t resist.
It appears that there are two white wires. One to the common and one to the master valve/pump start terminal. Is there a master valve installed in the system? Or a pump start relay? Make sure the right white wire is connected to the correct other half.
If a pump start relay is in place, does the pump come on when a zone is activated?
If a master valve is in place, does a zone work when the master valve is manually activated? If not, does a zone come on when both the zone solenoid and master valve solenoid are manually activated (i.e. checking to see that there is actual water pressure in the system).
When replacing the wires on the terminal, I’d also leave a lot less exposed copper - yes I know it is only 24 VAC, but I’m thinking of shorting out other components in the box.
Do you have access to a volt meter to see that there is still 24 VAC on the system (i.e. it didn’t short out when the wires were chewed)?
And I presume your dog is some sort of wire hair breed - again couldn’t resist.