I have a few notes on what may have been the 16th-, 17th-, and 18th-century understanding of penguin. Samuel Johnson's Dictionary of the English Language (1756) has the following entry:

PENGUIN. s. {anser magellanicus, Latin} 1. A bird, though he be no higher than a large goose, yet he weighs sometimes sixteen pounds. Grew. 2. A fruit very common in the West Indies, of a sharp acid flavour. Miller.

Though Johnson's Dictionary has no entry for auk, it does one for the word pinguid:

PINGUID. a. {pinguis, Latin} Fat ; unctuous. Mortimer.

Anser magellanicus ("Magellan goose") is a species name by which the great auk was sometimes known at that time, as this excerpt from John Hill, An History of Animals (1752) indicates:

The Penguin This is a very large and singular bird ; it is equal to the common goose in size, but in all respects resembles the species of Alca already described, except in the specific distinctions : the head is large, and flatted on the crown ; the beak is of a kind of triangular figure, compressed at the sides, and a little hooked just at the extremity ; it is between three and four inches in height, and has eight of the furrows which distinguish the birds of this genus on it, ... The head is black, only that there runs a white line on each side, from the beak to the eyes: the beak and wings, and indeed the whole upper part of the body are black, and the breast and belly, or whole under part are white : the wings are very short, and the tail also is short : the feet stand very backward : the legs are short and black, and the toes connected by a membrane. This is a native of most of the northern parts of Europe, and has been described by all the writers on birds. Willughby calls it Penguin nautis nostratibus dicta quae Grofugel Hoieri esse videtur ; Bartholine calls it Avis Garfahl ; Clusius, Anser Magellanicus ; Wormius, Anser Magellanicus Penguin ; and Ray, simply, Penguin. It feeds on sea-fish, and on many of the insects and small animals that frequent the shores.

This might seem to establish that in the mid-1700s the name penguin referred only to the great auk. But later ornithologists point out that there was considerable confusion in earlier times about not only the common name penguin but also the species name Anser magellanicus. As W. H. Mullens, "Notes on the Great Auk," in British Birds, volume 15 (1922) observes,

The first so-called "scientific" reference to the Great Auk as distinguished from those contained in early voyages and travels is that made by Clusius [in 1605]. On page 101 of his above-mentioned work, under the heading "Anser Magellanicus," Clusius gives the figure (here reproduced) and description of the South American Penguin. ... So much for the South American Penguin; it has been necessary to give the passage at length since we shall see other authors following Clusius have described the Great Auk under this title, i.e., Anser Magellanicus.

Clearly, the identification process didn't get off to a good start. Symington Grieve, The Great Auk or Garefowl (1885), devotes a full chapter to "Some Names by Which the Great Auk Has Been Known." The Internet Archive has posted a plain-text version (with some OCR mistakes) of the complete text of Grieve's book. Here is part of Grieve's section on the name penguin:

Penguin.—A name by which it [the great auk] was known in almost all the countries it inhabited during the last century was "The Penguin," or "le Grand Pingoin;" and there can be little doubt that this name, which appears to be of Welsh origin, and which is now given to a class of birds inhabiting the Southern Hemisphere, was originally given to the Great Auk, ... This confusion of the names led to no end of trouble among ornithologists, very few of whom fifty or sixty years ago really knew what were the points of difference between the species, and some were to be found ready to deny that such a bird as the Great Auk ever existed, ...

Grieve then quotes from volume 3 of Hakluyt's Voyages, George Peckham, "A true Report of the late discoueries and possession taken in the right of the Crowne of England of the Newfound Lands [in 1583], By that valiant and worthy Gentleman, Sir Humfrey Gilbert, Knight," chapter 3:

And it is very evident that the planting there shall in time right amply enlarge her Maiesties Territories and Dominions or (I might rather say), restore her to her Highnesse ancient right and interest in those Countries, into the which a noble and worthy personage, lineally descended from the blood royall borne in Wales, named Madock ap Owen Gwyneth, departing from the coast of England, about the yeere of our Lord God 1170, arrived and there planted himself and his colonies and afterward returned himself into England, leaving certaine of his people there, as appeareth in an ancient Welsh Chronicle, where he then gave to certaine islands, beastes and foules, sundry Welsh names, as the Island of Pengwin, which yet to this day beareth the same. There is likewise a foule in the saide countreys called by the same name at this day, and is as much to say in English, as white heads, and in truth the saide foules have white heads. There is also in those countries a fruit called Gwynethes, which is likewise a Welsh word.

Grieve next considers a mention of of "Penguin" in another account that appears in Hakluyt's Voyages—The Voyage of M. [Robert] Hore in 1536. That account includes this mention of "penguins":

From the time of their setting out from Gravesend [in April 1536], they were very long at sea, to witte, above two moneths, and never touched any land untill they came to part of the West Indies about Cape Briton [in what is now Newfoundland], shaping their course thence Northeastwardes, untill they came to the Island of Penguin, which is very full of rockes and stones, whereon they went and found it full of great foules white and gray, as big as geese, and they saw infinite numbers of their egges.

Robert Gray, "On Two Unrecorded Eggs of the Great Auk (Alca impennis) discovered in an Edinburgh Collection ; with remarks on the former existence of the bird in Newfoundland" (1887) notes the following comments by John Forster, in History of the Voyages and Discoveries Made in the North (1786) regarding Hore's voyage:

"A person of the name of Hore, says Forster, set sail in 1536 from London with two ships—the 'Trinity' and the 'Minion'—about the latter end of April. They arrived at Cape Briton, and from thence went to the north-eastward till they came to Penguin Island, an island situated on the southern coast of Newfoundland, and which was named after a kind of sea-fowl which the Spaniards and Portuguese called Penguins on account of their being so very fat, and which used to build their nests and to live in astonishing quantities on this little rock."

Grieve very briefly dismisses Forster's etymological analysis, which (Grieve says) is echoed by "Caroli's Chisius":

This derivation is one of the old-fashioned kind, and its absurdity does not need to be pointed out.

Finally, turning his attention to the species name Anser magellanicus, Grieve has this:

Anser Magellanicus, seu Pinguini.—Anser Magellanicus, seu Pinguini was the name used by Olaus Wormius in 1655 for the Great Auk; but he must have had a confused idea about the bird, for though he figures the Great Auk from one got from the Faröe Islands, and which he kept alive for several months ... yet he describes it as the Anser Magellanicus, seu Pinguini of Clusius, while that author describes the true Great Auk as the Mergus Americanus, and the penguin of the Southern Ocean as the Anser Magellanicus.

For almost 350 years, European ornithologists exhibited considerable confusion about precisely what bird or birds the word penguin referred to; but it looks as though etymologists may beat that record of confusion when it comes to identifying the origin and meaning of the word.