Interesting. Joining a circuit in Australia is via crosswind, downwind, base or final leg (conditions permitting) An inbound call should be made as "joining overhead" which is standard if circuit direction is unknown or there are many aircraft on the circuit. In some countries the call is "descending non-traffic side for rwy xx" which is equivilant to joining crosswind from overhead in Australia.
There is no such thing as JOINING early downwind for example. However it is, and has always been common practice to give a position report (if required) in the circuit of (for example) early/mid/ late downwind. As well as wide downwind,high downwind, wide left/right base or long final. These are all standard calls in aviation in many countries and if in a control zone these are terms ATC understand. Terms such as "long downwind" are non conventional and are not standard calls.