You probably have the PID programmed with the "outY" parameter set to "2". This is the mode where the main control output is set to drive an SSR. In this mode, the JLD612 has two alarm relays available to use.
J1 is a double throw relay for alarm1 functionality.
Pins 4 & 5 form a normally closed path (switch is ON when alarm1 is NOT active and OFF when alarm1 IS active).
Pins 5 & 6 for a normally opened path (switch is OFF when alarm1 is NOT active and ON when alarm1 IS active).
J2 is a single throw relay for alarm2 functionality.
Pins 13 & 14 form a normally open path (switch is OFF when alarm2 is NOT active and ON when alarm2 IS active).
All of these relay contacts are rated for 3A @ 120V. 3A @ 120V is 360 watts. You pump claims to be 10 watts, so a 3A relay is WAY more than sufficient to drive the pump. Hell, you could drive a March 809 pump with those relays if you wanted to.
You can use either alarm1 or alarm2 for this. It doesn't matter. I will give the example for using alarm1.
Connect the 120V hot line from your power source to pin 5 (you might want to put a 3A fuse on this line to protect the relay in the PID)
Connect the hot side of your pump receptacle to pin 6 of the PID.
Connect the neutral from your power source to the pump receptacle.
Connect the ground to the ground screw on the receptacle.
You have just set up the wiring so that when alarm1 triggers, the pump receptacle will turn on.
Now you just need to program the PID properly.
You are not using the main output of the PID to control this, so the normal set temperature you are used to using on the main display is irrelevant. You need to program the alarm1 parameters for this.
Press [SET] and enter the code "0001" on the PID to get to the alarm parameter menu. Press [SET] again. You will be able to scroll through 5 items now.
"Sv" = is the main control set temp. Irrelevant for you here.
"AH1" = temp value that will cause alarm1 to trigger
"AL1" = temp value that will cause alarm1 to stop
"AH2" = temp value that will cause alarm2 to trigger
"AL2" = temp value that will cause alarm2 to stop
Since we are using Alarm1, you want to mess with AH1 and AL1.
Let's say that you want to keep your water bath in the range of 66-68 degrees. Set AH1 to 68. Set AL2 to 66.
Save the settings by cycling to "End" and pressing [SET].
Done.
The PID is going to be trying to drive the SSR based on the reading from the probe, but you won't have the SSR feeding to anything here so it doesn't matter.
But, if the temp ever reaches 68 degrees, the J1 relay will activate and pins 5 and 6 of the PID will basically be connected together, which will allow 120V to pass through the J1 relay and reach your pump. The pump will come on and start circulating your ice water.
When the temp falls back down to 66 degrees, the J1 relay will deactivate and break the connection between pins 5 and 6, which will cause your pump to shut off.
If you followed all of that, you should be able to see how you could use the alarm functions of the PID to make a 2-stage controller. You just programmed Alarm1 to perform a basic ON/OFF cooling function. You could also program alarm2 to perform a basic ON/OFF heating function. Like... set AH2 to 64 and AL1 to 66 or whatever. Heating will now come on at 64 and stay on until 66.
Or you could use the existing SSR output to perform heating if you wanted to by using the normal set temp value and plugging a heater into the receptacle that the SSR is controlling.
Lots of options on that little PID. The alarm relays are super flexible.
If that didn't make sense, let me know. You will have to do some re-wiring here to make this work, but in the end you can have it set up so that one outlet on the box is for something you want the SSR to control and another outlet is for something you want the alarm to control.