How is an after-market rear camera trigger connected?

Aftermarket Backup cameras normally pickup their trigger from reverse lights. If so, that wire also provides 12 volt power to the camera. So I am a bit confused as to how your camera can work if that extra wire is indeed its trigger/power lead and is not connected.
Best thing to do would be to trace the camera wires starting at the camera and work backwards. There should be one single cable out of the camera which then splits into two sets. One for power (positive and ground) which goes to tail light, and the other for signals going to the display in the head unit.

 

All cameras have 3 wires _+ RCA output.

Ground
12VDC
Reverse

The reverse signal can be picked up at the dash - or as I did form the tail lights. Since you were playing with them - likely that's the issue. There should be a wire that taps into the reverse light on one side or the other - that probably came loose.

 

The backup camera I installed on my 2010 had RCA video and two wires -- 12V and ground. Some people run the 12V to the reverse light, so the camera is only powered when needed. I ran mine to a constant 12V. But I don't understand how yours works only sometimes, unless the wire from the reverse light goes to the head unit, and not the camera.

I think your best bet is to take it to the ship that installed the backup camera. It's most likely supposed to connect to the backup light, but if you send 12V to ground you could fry a relay or worse; it's not worth it.