This Tahitian parakeet is one of my favorite birds, unfortunately it doesn’t exist any longer because it was wiped out by introduced cats, dogs and rats.
Here are two depictions that were both made in the year 1792, when the HMS Providence stayed at the island of Tahiti with the mission to collect breadfruit trees and other botanical specimens from the Pacific to be transported to the West Indies.
Miller’s Rail is one of the more commonly known so-called mysterious birds.
This species is actually known exclusively from a single drawing made by Georg Forster sometimes between 1772 and 75 during the second voyage of James Cook [and a copy of it made by John Frederick Miller, who described the bird as a new species in 1784]. The annotation just states that it is a Rallus minutus, [a small rail], [called] Maho, [and coming from] Taheitee, [Tahiti].
The drawing is rather a crude one, not „fieldguide-suitable“ and shows a small bird, clearly identifiable as a crake, with rather dark, almost black feathers, sitting on its red legs.
The bird could very well just be a Spotless Crake (Zapornia tabuensis (Gmelin)), which today is still [patchily these days] distributed all over Polynesia, and of course was even more so 250 years ago!
There is yet another quite detailed description supposed to be of this species, made by John Latham in 1785 from the actual type, that is now lost.:
LENGTH six inches. Bill three quarters of an inch, black: the head, neck, and all the under parts of the body, dark ash-colour: palest on the chin: the upper parts, and wing coverts, deep red brown: quills dusky, edged with white: edge of the wing, and the first quill feather, white: tail an inch and a half long, rounded in shape, and black: legs dusky yellow. Claws black. Inhabits Otaheite, and the Friendly Isles. Sir Joseph Banks.“ 
The same book contains the description of a variety of the Tabuan rail [now Spottless Crake (Zapornia tabuensis)] from the island of Tanna in the Solomon Islands chain which is often regarded to as being the description of the actual type specimen of Miller’s Rail, however, the description differs quite significantly from G. Forster’s depiction.:
“This varies in having the plumage more inclined to brown: the vent white, transversely barred with black lines: legs red. Inhabits the island of Tanna. Sir Joseph Banks.” 
The island of Tanna, mentioned here as place of origin of this bird, was just one of several islands that were visited by Cook and his entourage in the middle of the 18th century, and the place names given by J. Latham are very often completely wrong, however, the descriptions on the other hand are rather complete and trustworthy.
It has to be taken into account that such old books most often lack any kind of register and that they mostly just use common names but lack scientific ones, searching inside them is a long-term venture.
It is now quite well known that in former times probably all of the islands in the tropical Pacific Ocean were inhabited by endemic rails, with several islands being known to have been inhabited by more than one species, and in many cases these were congeneric species, meaning species from one and the same genus – something that today is extremely rare, which, however, is a relict situation, left behind by human-induced extictions. 
 John Latham: A General Synopsis of Birds 3(1): 235. Leigh & Sotheby, London 1785  John Latham: A General Synopsis of Birds 3(1): 236. Leigh & Sotheby, London 1785  David W. Steadman: Extinction and Biogeography of Tropical Pacific Birds. University of Chicago Press 2006