On the 3 pin, one set (perhaps pin 1 and 2) is probably normally closed (NC and a Common), while with the other set of pins (perhaps pin 2 and 3), would be normally open (NO and Common).
If this is the case, you can check it with an ohms meter for sure, and when the dry contact is triggered, see if one set goes open, and between the other set would go closed.
I'm just guessing here, but that is what I suspect.
The manual might have more info about the pins/pinout on the inverter (I've seen it specified in some inverter manuals I've looked at before, including my MPP Solar inverters, the manual shows the pins).
So if the generator starts when you close the contacts, you'd only use the 2 pins on the inverter which go closed (NO) when the dry contact is engaged, and when it's released it should make the circuit open and shut off the generator.
Example showing my MPP Solar LV6548 dry contacts pinout: