A Dermatologist in Australia is distinguished from other medical practitioners by having the qualification of "Fellow of Australasian College of Dermatologists" or FACD.
Skin cancer clinics are staffed by general practitioners in general. At times, you may come across the confusing sign of "Specialist Skin Cancer Clinic." The "specialist" running these skin cancer clinic are usually surgeons with NO specialist training in dermatology.
"Cosmetic Physicians" are also general practitioners with NO specialist training in dermatology. These "Cosmetic Physicians" usually packaged themselves as "Skin Cancer Physicians" as well. They are neither Dermatologist nor Plastic Surgeons.
One may also come across practitioners with qualifications such as being members of some "colleges" of cosmetic medicine or having Diplomas in "Practical" or "Clinical" Dermatology. These are NOT specialist qualifications. The training associated with these qualifications is aimed at General Practitioners, usually involved a fee-paying course of short (days to weeks) duration with the major part of training done by correspondence.
To check your Dermatologist''s credentials, you could check the web site of the Australasian College of Dermatologists or simply go to the Medical Board of Australia. Please note that there is an entity known as the "Medical Register of Australia"; this is simply a commercial advertising publication similar to the Yellow Pages.