A lot of people tend to work on their moped alone and often get fooled by what is really going on with the clutch, belt, and front variator because they cant see a side view of the belt and variator because they are standing over and pedaling.
What your describing is the the chattering of the starter shoes working fine and what might be happening is the belt is slipping around the front variator it what I call "tiny loops" each time a loop goes around the front variator it throws little slack spans in the belt which transfer back to the rear making it chatter back and forth.
Try tightening your belt a little.
Your shoes ar installed correct. Do not ever put any grease on the shoe linkage or pivots. It will never help, is not needed and will eventually get on the run face of the bell.