The truth is that there is interaction, even an interconnectedness, between local government and counterparts at state and federal level. Canberra itself may have to make do with honorary mayors but it plays host to Lord Mayors and their vanity through patronage at the Commonwealth level.
In recognition of the fact that Newcastle was New South Wales' second largest and oldest city, council applied to have a Lord Mayor, which was granted by King George VI
Newcastle became the first city that was not a capital to have a Lord Mayor. I'll post a Newcastle ale soon but with selective hyperlinks since there are rather a lot of them when you count the mayors up to 1947.
The other cities with Lord Mayors are Wollongong and Parramatta.