Hi @iamdago, how did you try to get in touch with me? Did not mean to ignore you, but as @DLane has mentioned, I do not have time to read every new post lately. Be sure to include “@” in front of whoever you wish to see your post, @Gene for example, otherwise it will not be very visible for them.
I recommend Rachio 2 or 3, as “e” version is missing flex daily schedule which is personally my favorite.
You are free to explore other controllers, but few will be able to comment about any particular system outside of what we are familiar with.
Start relay you’ve linked to will work, looks to be the same version as sold via homedepot / lowes, but with unrelated photos in the listing. Be sure to check other sellers, if you prefer to use prime shipping.
You look to have 220V (two pink wires) and Netral / GND (white wire) running to your old Intermatic.
Ideally you’d run separate 110V to the Rachio enclosure, as @DLane has mentioned, but you can also use either of the pink wires with the white wire to get 110V. Your pump & start relay should still use both pink wires for the full 220V.
Be sure to buy a multi-meter, from the DIY savings, in order to verify all of the voltages and turn off the power at the breaker before working with any of the wires.
The valve manifold that @Dlane has linked to in #4 of his responce is great if you wish to bury your valves, but with just two zones, it may be easier for you to keep the valves above ground and install a pair of above ground valves, such as these from Orbit (link), again be sure to check “other sellers” for prime delivery. On that note, please verify if you have 1 inch plumbing or 3/4 and buy an appropriate valve.
Above ground valves come with the anti-siphon valve (back-flow preventer), which you strictly do not need but there will not be any negative side effects from having them.
For anything else I’ll refer you back to @Dlane’s comment, he did a great job outlining the needed parts.
Most of the important questions will come as you get stared. Do not hesitate to ask and you should have a straightforward conversion.
Some guy Gene (better late than never)
P.S. Are you sure you have two zones? There are six pins on the timer, indicating that you likely have three zones. (three pins to turn ON the zones, and three to turn them back off).