You can just do a Voltage test on thin wire terminal, where the thin wire connects to the starter Solenoid / "relay". Just see if you get a Voltage reading at that terminal when you press the starter button.
(On my scooter I need to loosen the Solenoid / relay to get access to this specific terminal)
If you get a Voltage reading at the small terminal when you press the starter button, the solenoid / relay should be faulty.
If you don't get a Voltage reading at the small terminal when you press the starter button, the starter Switch (at the button), brake Switches (at the levers), or Ignition Switch could be faulty.
My scooter has two fuses inside the battery box. If one of those two fuses are blown - the scooter doesn't start. So maybe you should also check for a blown fuse inside the starter box - or a blown fuse elsewhere.