Did Shakespeare really coin "Alligator"?
Solution 1:
In a word, no. Shakespeare did not coin 'alligator', not even in the form 'aligarta' (The most excellent Tragedie of Romeo and Iuliet, 1597: "And in the same an Aligarta hangs"). What evidence there is suggests, rather, that he (or whoever produced the 1597 text) might have adopted and singularized the form after its use by Walter Bigges (died 1586) in the 1589 A summarie and true discourse of Sir Francis Drakes VVest Indian voyage: "…where we killed also many Aligartas aforsaid".
Note that the OED etymological history of 'alligator' in the Third Edition, September 2012, is (by my measure) greatly changed from that in the Second Edition.
Etymology: < Spanish el lagarto < el the + lagarto lizard (13th cent.), kind of large New World reptile, alligator (first half of the 16th cent., no longer used in this sense), ultimately < classical Latin lacerta …, perhaps via an unattested post-classical Latin form *lacarta.
Compare ( < English) French alligator (1663; > scientific Latin Alligator, genus name (1807 in Cuvier)). Compare also ( < French) Spanish aligátor (1797 or earlier; the more common word is now caimán…).
The remodelling of the ending of the word from -arto or -arta to -ator or -ater probably results from association with agent nouns in -ator or -ater….
The textual evidence cited by OED in the Third Edition has changed. Saliently, the 1568 quotation in the Second Edition has been omitted, and a 1555 quotation "relevant to the development of a sense but not directly illustrative of it" stands in its stead.
The problematic 1577 quotation in the Second Edition remains, although the nature of the problem receives an editorial tip of the hat in the Third Edition it did not feature in the Second Edition: "… [Sp. Caymanes, que llaman Lagartos].]" The second, dangling, close-bracket may indicate that a typographical error failed to enclose the entire attestation in brackets showing that the use of 'lagartos' in the quote was not considered directly illustrative of English, as opposed to Spanish, use of the word.
The 1589 use I cited from Bigges, the singular form of which parallels the plural form found in the 1597 Shakespeare, does not get a mention in the OED Second or Third Editions, but the 1591 Knivet attestation from the Second Edition does appear in the Third; the next attestation following the Knivet quote in the Third Edition is from the 1597 publication of Romeo and Juliet.
I find the note in the OED Third Edition regarding the remodeling of the ending of the word far more palatable than the Second Edition's "whence by popular corruption" and following convolutions, if for no other reason than Occam's razor.