The 'black stick' is the center spindle on which the 'drive dog' would be sitting, if it hadn't been removed at an earlier time. A head without the drive dog is only going to spray a full circle. If you remove the cover from a head that's spraying a part circle, you'll see what you're looking for. With any luck, another of the full-circle heads you have will still have the part.
By the way, when a full circle head is no longer wanted, this hints at changes that may make preserving the original heads not worth the effort.