Saint Paul Duck (Anas sp. ‚Île Saint-Paul‘)
While describing a new extinct bird, the Amsterdam Island Wigeon (Anas marecula) in 1996, the two authors Storrs L. Olson and Pierre Jouventin mentioned an account from the middle of the 18th century, and quoted another author, W. R. P. Bourne.:
„Of far greater interest, however, is the report of the explorer John Barrow, who was on St. Paul Island on 2 February 1793, where he mentioned the presence of „a small brown duck, not much larger than a thrush“ that was „the favorite food of the five sealers living on the island“ (quotes brom Bourne et al. 1983).“ 
To me, it seems, the two authors did not actually check the original source, John Barrow, here.
Well, but I did …! 😉
By the way; another well-known author of bird/extinction-related books, Julian P. Hume (in Extinct Birds; in the 2012 – or in the 2017 edition), even gives a completely wrong source.:
„John Barrow: Some Account of the Public Life, and a Selection from the Unpublished Writings, of the Earl of Macartney. Amsterdam London: T. Cadell and W. Davies. 1806“
I checked that source too! 😉
But back to the actual source, what does it really say?
„On the 1st of February we discovered the two islands of St. Paul and Amsterdam, and on the evening of the same day anchored on the eastern side of the latter, at the distance of about a mile from the shore. …“ 
The latter one is Amsterdam Island, right?
What follows are several descriptions of the island, of its geology, and some quite interesting philosophical reflections about the fact that some parts of the planet appear to be older/younger than others, and that islands apparently can just appear out of nothing or disappear into nothing, obviously without aid of an unearthly higher being (remember; Darwin’s ‚On the Origin of Species‘ first appeared 66 years after).
… oh, and a list of birds of course.:
„The number of birds was likewise astonishing, and the two causeways were strewed with teir eggs. During our short stay on shore we obtained the following birds:
Anas, A small brown Duck, not much larger than a thrush, and apparently not described by naturalists.“ 
The author still speaks about Amsterdam Island here, so this is the Amsterdam Island Duck (Anas marecula Olson & Jouventin)!
Its this little passage – „A small brown Duck, not much larger than a thrush“ – that apparently was copied again and again by several authors without checking the original source.
On page 155 resp. 156, the author reports about five seal hunters, that „all lived in a small miserable hut, as dirty and offensive as that of an Hottentot; and it was surrounded on every side by the dead carcasses of seals and sea-lions.“ 
„If the smoke and the fires of Amsterdam Island had excited our curiosity, the discovery of two or three human being running along the shore, as our ships approached it, on so miserable a spot, and so distant from any other land except the little neighboring island of St. Paul, caused a still greater degree of astonishment. …“ 
This passage clearly still refers to Amsterdam Island!
„The birds, they observed, had a strong fishy taste, to which, however, long habit had reconciled them: those that were the least so were the blue petrel and the little brown duck.“ 
There again, the small duck from Amsterdam Island!
The Amsterdam Island Wigeon clearly survived into the 18th century, since it is clearly that bird that is mentioned in the so often (incorrectly) cited quotes. There may a duck have existed on the Île Saint-Paul, however, up to now there is no proof for that assumption!
 John Barrow: A voyage to Cochinchina, in the years 1792 and 1793. To which is annexed an account of a journey made in the years 1801 and 1802, to the residence of the chief of the Booshuana nation. London: printed for T. Cadell and W. Davies 1806
 W. R. P. Bourne; A. C. F. David; C. Jouanin: Probable Garganey on St. Paul and Amsterdam Islands, Indian Ocean. Wildfowl 34: 127-129. 1983
 Storrs L. Olson; Pierre Jouventin: A new species of small flightless duck from Amsterdam Island, southern Indian Ocean (Anatidae: Anas). The Condor 98(1): 1-9. 1996