I think your confusion comes from people discussing definitions (the 'end result') without explaining the foundations of why the definitions are so (the reasoning):
In the Australian constitution there are three sources or law making/enforcing powers, that is the judiciary (courts - interpret and have the final say on questions of law, that is how the law is applied), legislature (parliament, makes the law 'supreme law making power') and the executive (in theory headed by the Crown with the power vested in the governor-general but in practice vested in the queen, administers the law using some powers it has simply due to history [that is, powers the British monarch had and therefore is exercised by the executive and the crown's representative] and some granted by parliament).
In the constitution there is also a separation of powers, that is, the three powers are supposed to have a degree of separation from each other (for several reasons including acting as a check and balance on each other).
The cabinet is a part of the legislative power, that is, the cabinet is made up of the prime minister and the senior ministers in parliament. The cabinet is responsible for most of the legislation brought to parliament, that is because the cabinet determines the party policies that would be pushed in parliament. The cabinet is not in the constitution, but is rather something that has developed in practice due to the way party politics works in Australia.
The executive council, is made up of the governor-general (representing the Crown) as well as the prime-minister and senior ministers. The executive council is responsible for administering the law, that is it can make regulations under powers delegated to it by parliament (such as road rules). Each minister in the executive council is responsible for a different portfolio, that is an area of executive power which is exercised by a government department, for example a health minister would be in charge of medicare and minister for police would be in charge of the Australian Federal Police in the federal context. The executive council also has some other powers, such as having the final say on matters of war and peace. The executive council sits at the top of the executive power. The executive power includes, as other posters have mentioned, several law enforcing/making bodies, such as the police, as the police also 'administer' the law.
In essence, although the cabinet and executive council are made up of mostly the same people, both the entities have different purposes and roles in the Australian governance and and are a part of different powers granted in the Australian constitution.