(b) That's right. In modern Australian law, it is recognised that a company can be used to run a small, one-person business. This was not always the case. Company law initially developed with much larger enterprises in mind, and it was only in 1897 that the courts recognised that the full benefits of a company's separate legal existence should be extended to small businesses as well. This view is now reflected in the relevant legislation.
For general business purposes, a company limited by shares is appropriate. In current law, a distinction is drawn between 'proprietary companies', and 'public companies'. A proprietary company is not allowed to have more than 50 'members' (ie persons who have been allocated 'shares' in the company), not counting shareholders who are also employees. And a proprietary company is not allowed to raise funds for its business by offering shares to the general public. If a company wants to have more than 50 non-employee shareholders and offer shares to the general public, it must be a public company.
Proprietary companies are required to include the abbreviation Pty in their company name, to indicate the type of company they are.