The two small connections circled in light purple are connected to each other. One is male and other is female. Larger connection in bottom right goes behind the OBD II port.
The button is a push to talk button for voice command and is optional. We don't have a video for a non-sync vehicle, but the installation is the same regardless of the model of vehicle.
The diagram is pretty straight forward. The pink harness and the green harness in the diagram have the same connections, so they CAN be plugged into the wrong place and result in blowing a fuse, or even worse, torching the ACM.
The one that's tagged for the security gateway CANNOT be plugged to the ACM, as it can cause reverse polarity. They need to disconnect the harness from the back of the OBDII port, and put this passthrough in there, and plug the connector that used to be at the back of the OBDII port into the back of the harness.
Please verify they did NOT do this backwards, because if they did, they need to check their fuses and hope the ACM didn't get fried. As for the display not powering on, did they make sure that the Sync harness connector was fully seated to the back of the APIM?