An air carrier pilot briefed the complex RDM engine out procedure but failed to brief a normal departure. After takeoff; the crew failed to turn and could not find the transponder identification button to positively identify themselves.

Narrative

I was the Captain of a flight from RDM to ZZZ. During the pre-departure briefing; I discussed at length the rather convoluted special engine-out procedure for Runway 22. Somehow; we failed to clearly discuss the 'normal' takeoff procedure; which was from the ATC clearance to turn directly to the DSD VOR; then on course. I was Pilot Flying and mistakenly assumed we were to fly runway heading. We did not make the immediate right turn to DSD; and adding to the confusion was ATC asking for an ID. As our cockpit equipment are all very nonstandard from one to another aircraft; it took a few seconds to hunt for the ID button on the transponder; while we should have been making the turn. We were asked by ATC to 'say position.' Before I could react; my First Officer reported over 'DSD;' when in fact we were approximately 5 NM south-southwest of the station. Then; when radar contact was established; ATC was understandably confused at our non compliance and erroneous position reporting. I was given a phone number to contact after landing; which I did; and discussed what happened and how to avoid non compliance in mountainous terrain in the future. I am reminded to ensure and require complete preflight briefings from my First Officers and the importance of maintaining vigilance in an especially diverse fleet of aircraft.