Beaver:
You are absolutely amazing. Tell me which of my examples do not fit into your definition? Are you telling me Japan Airlines' JFK-San Paulo is not a fifth freedom route, or Air New Zealand's LAX-LHR is not one? The last time I checked, the US and Brazil are not a part of Japan. Also, the US and UK are not part of New Zealand. The same can be said for all my examples.
In fact, your definition is incomplete. Fifth freedom rights have to be flights that originate or terminate in one's own country (but it does not have to be using the same aircraft, as one might recall UA 1/2 changed aircraft in HKG and LHR). That is, Air New Zealand's LAX-LHR flight has to be a sector of a flight that originated in New Zealand. Indeed, the complete itinerary of the said flight is AKL-LAX-LHR. If NZ can operate an independent flight between LAX-LHR, then it's not a beyond or fifth-freedom right. FYI, it would be called a seventh-freedom right. For example, the US agreements with Macau and the Philippines allow seventh-freedom rights for UPS and FedEx.
Just in case you don't know, Cabotage is also known as the eighth freedom. And to complete the whole thing, the sixth freedom is to carry passengers from Nation A to Nation B via one's own country. An example would be Air Canada carries passengers from Europe to the US via Canada.