Camera external trigger; What is the shutter signal?
Unfortunately I only rarely have access to the camera

You don't need the camera to diagnose this. Just use an ohmmeter to identify which switch terminals are shorted when the button is pressed. It may connect to one terminal when half depressed, and also to another one when fully depressed. The camera may require that both switch contacts be closed in sequence to operate the shutter.
Once you have determined which switch terminals need to be shorted, 'press' the switch from the Arduino using small PhotoMOS relays wired to it. These have the advantage that you don't have to worry about output polarity, so even if the camera does funky stuff on the trigger wires it should work the same as if you were pressing the actual button.