Let me ask this since you were told to use a specific type stud.

Do those studs screw in to the hub then instead of being pulled through the backside? Or did the studs you have pictured just snap off?

Now if the studs snapped off chances are the studs were too long to begin with. Now if the lug nuts were coming loose they could have been bottoming out and not giving enough clamping force to hold the wheel tight against the hub as it should. This will cause the holes in the wheel to get screwed up and can cause studs to break also.

The lug nuts you have are shank style, just with a tapered seat. The shank part is supposed to help center the wheel on the hub.