Alas, our theory held true, both of the supplies you’ve tried are direct current version, DC is in the name 24vDC.
Here is the version that @MDNZ has found, which was verified to work (note the name: 24vAC):
Can you please take a picture of the transformer you’ve used? It may be useful for future reference.
Alas, I can’t recommend anyone who can do soldering in NZ, being from USA myself. You may wish to inquire within stores which fix phones / computers.
How many zones does your rachio and home irrigation system have? Do you have any spare zones available on rachio?