Rocky, that exception is for private cars.
You can take a child in a private car without a car seat if it's short, necessary and unexpected. Over 3 years old the child must use a seatbelt instead in this case. Under 3 years they don't have to.
So for example if you are in town and see someone you know and offer a lift home, that's short and unexpected, but not necessary = no exception. If it was pouring down with rain and you have a small child with no rain cover then maybe necessary and that's fine.
If you need to take your child to hospital, that is necessary and unexpected, but you have to choose the closest - if you decided to drive right across the city to a different hospital for no reason, that's not short = no exception.
If you regularly drive a short distance so that they can attend school, that is short and necessary, but it's not unexpected. You have to use a car seat.
In reality of course nobody is monitoring 24/7 to make sure you're using a car seat but if you were stopped by police for not using one, you can use the "short, necessary, unexpected" defence to explain why you are not using a car seat even though the child is of the age where they need one.
The other exception is if you have three children, a five seater car and can't fit a car seat in the middle, the oldest child is allowed to travel without one if they are over 3.
(And medical exemptions).
It's never legally required in taxis although it is still safer to use one obviously.
If you need to put a child on your lap always put them on top of your seatbelt. If you are both under your seatbelt, your weight could crush them against the belt.