(b) No, that's wrong. A proprietor's name on its own usually says nothing about the type of business being conducted. It is often desirable to include some description of the business in the name. For example, Edward might think it an advantage to trade as "Delgardo Plumbing" or "ED Plumbing".
If the persons who operate a business use a business name that consists of anything apart from their own name, the Business Names Registration Act 2011 (Cth) requires that the business name be registered in a register of business names. This requirement applies to different types of business organisations, such as sole traders, partnerships and joint ventures. A corporation can register a business name, but a trust cannot unless it is incorporated.
Once trading begins under particular name, the reputation of the business will come to be associated with that name. It is important to choose a name that will remain appropriate as the business develops and expands, even if that name initially requires registration.