There is a species of Crayfish in Australia with three common names, the one used most commonly by the public and most associated with the species (including by a museum) is not registered in the Australian Faunal Dictionary.
The other two were used in the original description and subsequent revisions (occasionally switching with each other). They are not used by the public (unsure about professionals, but they are more likely to use scientific name) but are registered to the AFD.
Which name takes priority, the widely accepted and used one, or the older two listed by the AFD?
There is no set of rules governing which names should be prioritized for a given location. Curators can reprioritize names as needed, or, if it isn’t clear, a curator could raise a flag on a taxon and invite other users to weigh in.
iNat defaults to the first name added taking priority, but curators can change this, this kind of change should generally involve discussion on a flag first. The name should be whatever the consensus on the flag is, iNat has no rule for which name to use
In general the most used name is prioritized, but that is not required, and different name might be prioritized if that name is misleading or otherwise problematic.