^^^ Mostly correct.
1. If you have Third Party Property Damage up to Full Comp you can lodge a claim (most insurer's come with $5000 to repair your vehicle if uninsured driver causes dmg - with TP and Fire & Theft products), Comprehensive has no limits aside from your sum insured. If they are uninsured they are f'd.
2. If the TP (Third Party) has insurance but you don't you will need to get them to lodge a claim. If they do not no amount of chasing his insurer will do anything as they are legally bound to take no action until he lodges a claim and puts your name to it. Only option you have if they refuse to lodge a claim is seek independant legal advise.
The above is given to you with general advise and may or may not apply to the products you hold, if you have any questions or concerns with what product you have, please consult your insurer.