Steve, think your picture is the correct thing. (if you open it up it is full of electronic). The plug you can check if it ihas connection on ALL 4 terminal, hence 4 pin flasher and socket will work is a (old) standard in europe
And yes the flascher internal connect the pin to ground. R is going to fuse = + bat. If I am correct this pin going to thhe buld is just an internal "duplicate" connetion to the "output" pin going the the adequate light bulbs left or right. -So doubble check from any + bat and a needle in the orange wire for contact as input for a volt meterOR just a spare bulb..set either flash light on /ignion on..etc..and you fence the real problem in. WELL it also tells you if the tach bulb is broken. 
I had some problems in "flashing" neither right nor left. Did buy a new now fully electronic flasher - guess same as yours above - from Hella ( Have it a home currently skiing). My issue was the broken "hazard" switch in the middle of the diagram above, as one internal contact was broken (BMW 2002 part 100€)
So there are cheap flascher from Hella or Bosch. (< 10 USD plus ship ..sorry my wife just left to FL) .get you the part number Wednesday--- if yours is not working.
Mat...from the ski slopes.