There are some subspecialties such as Consultant Neuropsychiatrist, which refer to Psychiatrists who have extra training in Neurology. Its important to note that Psychiatrists can prescribe medications to their patients, however Clinical Psychologists can not. The use of the title "Dr" can be confusing and traditionally it refers to a person holding a MBBS or PhD.
The national body (Australian Health Practitioners Regulation Agency) have largely de-regulated who is entitle to use the term "Dr" or "Doctor" which was previously legistlated by state law. It is now possible for many health professionals to use the title Dr without holding a Medical Degree or PhD (which were previously required alongside specific professions). The clinicians at Cygnet Clinic only use the title if they are doctors or hold a doctoral degree from an accreddited Australian university in accordance with the traditional use of the title. Other titles such as A/Prof (Associate Professor) are academic titles referring to a position within the heirarchy of university scholarship.
The use of Doctor as a noun is traditionally reserved for holders of MBBS such a Psychiatrists.