Cpdlc/ads
If you log on to CPDLC/ADS 15 mins prior to the boundary and still in voice contact, I make a voice position report and include the words 'CPDLC Established'. There is no right or wrong way to do it.
At the boundary, make a manual CPDL report. Check and make sure that you have an ADS connection. If it is working correctly, you won't have to provide another manual report.
And its true...the Aussies are a touchy lot, for whatever reason. If operating in Oakland area, it is straight forward. With the advent of auto reporting, especially in the OAK FIR, it virtually leaves a pilot with nothing to do but monitor the airplane. Occasionally on the B777, the 'Pilot Response' used to come up on EICAS because everything was working so smoothly.