Hi,
Sounds like a good project.
I wouldn't worry too much about how it operated in the past. The schematic is useful to show us what the machine has. I would rip all the existing wiring out and start from scratch. This was you can set it up and control it as you wish, even adding more functionality to the machine that it may never have had in the first place (such as programmable pre-infusion and wait times, pressure profiling, shots based on time, etc)
One thing I would definitely add is a PID to control the boiler temp rather than the existing thermostat. Huge upgrade!
It has a probe in the boiler to sense when it needs refilled. If this probe can be used in your Arduino design that would be good. It refills the boiler if using the steam wand (no pump active) depleting the boiler of water. Theres a few ways around this but using the probe would be ideal.
I've started a thread here about my project as I've got all the hardware setup and also working in Arduino, however, I'm currently looking at the best way of coding the whole menu-driven system.
https://forum.arduino.cc/index.php?topic=598777.0
Here's some photo's of my project so far:
https://drive.google.com/folderview?id=1B3v_CZkXpclEsZ6LjCiI0q16ndA4oxRd
I'll be looking forward to see how you get on with it, good luck!