(b) That's right. A business name is the name under which a person conducts a business in Australia. The use of an appropriate business name allows everybody to know who they are dealing with.
The use of business names is regulated by legislation. The Business Names Registration Act 2011 (Cth) requires business names to be registered in some circumstances, but not all.
Generally, persons are entitled to use their own names as their business name without any need to register that name. For example, if Edward operates his plumbing business as "Edward Delgardo" or "E Delgardo", he would not need to register that name as a business name.
Even in circumstances where registration is not actually required by law, having a registered business name often makes it easier for a business to secure banking and other services.