Door handle button plunger?
Yes, that's it.
I have found that the reproduction door handles look nice, but usually seem very gritty or stiff and hard to operate compared to originals. I'm going to have to take one apart and see if I can polish and lube it a bit to make them operate more smoothly.
The knurl on the pin is not used as a spline like on a window regulator shaft, or as a increased diameter for a tight fit like in a door hinge pin. In the door handle it is used as an increased diameter simply to keep the pin from sliding out of the assembly and to allow the spring and a plate to retract it.
Now, do you also know what the formed wire part is that I used for a clue?