As a property owner, you are responsible for all plumbing, including the pipes, fixtures and fittings, from the water meter, irrespective of whether the meter is located within or outside your property boundary.
Unitywater owns and is responsible for the infrastructure up to and including the water meter. In some instances, the water meter may be located within your property boundary, however, it is still the responsibility of Unitywater.
Council owns and is responsible for the stormwater pipes (the pipes that take away rain-water) outside your property. They are also responsible for private plumbing and drainage compliance.