The pH sensor board you linked to is biased at +2.5V with unity gain on the OpAmp. It should work directly with your probe.
The circuit I designed last year uses different voltages, as it's designed around standard pH probes (so +/-700 mV), trying to make the most of the ADC range. The circuit works, I built it on a piece of protoboard but never continued working on it. I'm quite sure it'll work a lot better on a properly designed board as I had a really hard time soldering the PCB connector to it. The OpAmp comes in in SSOP8 package, which I soldered on a breakout board, that plugged into a DIP socket - also not exactly the most ideal way of handling that.
If you do not understand that circuit, there's little chance you can get it to work well, let alone make the modifications needed for your +/-2000 mV probe. You have to adjust the bias to 2.495V (so no voltage divider there), and change the output OpAmp circuit to unity gain.
For better sensing, I've played around (but never built) with redesigning the circuit to produce a 0-1V output, as this can be read using the more stable built-in reference. Another idea I had is to connect the 2.495V reference to AREF of the processor, and then bring the output voltage into that range. Both ways you can greatly stabilise the reading you get in the Arduino. Or have an ATtiny read the voltage and interface with that over I2C. That's my ultimate pH sensor, if only because that'll work with both 3.3V and 5V power.
Then there's just that issue of ground loops left to deal with...