From what I researched the two on the left are the correct terminals for Chamberlain. If your wall button works fine then I suspect there is no good contact for Garadget wires at the opener’s terminal.
You can pull out the wall button wires from the opener’s terminal by pressing the tab below, then match the length of the copper on the garadget wire with the old wire, the put them back together side-by-side, maybe twisted for good measure.
There’s often additional circuitry in the wall button for extra functions (like locking the radio out and controlling the light), but good old short of the wires is normally reserved for the door motion. The duration of the short is configurable in Garadget’s settings.