Schedule 2 Part 1 of the Reform Act (Item 5-15 Page 152) refers to the definition of external administrators:
5-20 Meaning of external administrator of a company
A person is an external administrator of a company if the person is:
(a) the administrator of the company; or
(b) the administrator under a deed of company arrangement that has been entered into in relation to the company; or
(c) the liquidator of the company; or
(d) the provisional liquidator of the company.
Our reading of this item does not change the way appointees describe themselves. The Reform Act definition serves to group the appointments types so to conveniently refer to external administrators as a collective for convenience.
The appointment is still made under the particular section of the Corporations Act 2001 which covers separate appointment methods for each of the external administrators separately. For example a liquidator will still be referred to as a liquidator, etc.