I think it would be good if “reintroduced” and “self-introduced” categories were added to the establishment means for iNat. “Reintroduced” could be symbolized by a blue plus sign while “self-introduced” could be symbolized by an orange plus sign (much like how endemic and introduced are symbolized by a green star and a pink exclamation mark).
“Reintroduced” could be used when an animal is either purposefully or accidentally reintroduced to an area where humans previously wiped out it or a close relative. Examples would be elk and red wolf in the eastern US, wolves in western and central US, certain groups of wild horses and cattle in Europe (i.e. those at Oostvaardersplassen preserve in the Netherlands), black swan in New Zealand, yellow-crowned night heron in Bermuda, scimitar oryx in northern Africa, etc. Of course, there is some gray area in here, such as Aldabra giant tortoises in the Mascarene (the extinct Mascarene tortoises belonged to a completely different genus, so would this count as reintroduction?) and feral horses in the United States (there’s already a huge controversy over whether or not it counts as reintroduction).
Self-introduced refers to documented cases of an animal establishing populations by itself in a new area. Examples would be cattle egret in the Americas, glossy ibis in the United States, Eurasian collared-dove throughout Europe, green iguanas on Anguilla, and silvereye in New Zealand. Of course, there are still some gray cases here too (the monarch butterfly most likely spread throughout the Pacific by itself, but this was mainly due to humans introducing invasive milkweed to many Pacific islands, so would this really be natural?).
I think that including these two categories as establishment means will help to paint a finer portrayal of the complex nature of species establishment.