As already discussed elsewhere on this site, Sir Thomas Wyatt (1503-1542) and Henry Howard, Earl of Surrey (1516-1547) introduced the sonnet into English literature. While doing so, they also introduced a few changes, which are probably due to
- the lower number of rhyming words in English,
- a tendency towards pointed arguments, as exemplified earlier in Chaucer's "rhyme royal" (Kurt Tetzeli von Rosador, in Schabert 2009, page 572).
Wyatt introduced the rhyming couplet at the end of the sonnet; Surrey replaced the Petrarcan sequence of octave and sestet with three quatrains that use the rhyme scheme ABAB CDCD EFEF. The "volta" now came after the third quatrain instead of after the second. Wyatt and Surrey introduced the sonnet and their innovations during the reign of King Henry VIII, so the English sonnet form predates the Elizabethan era.
Shakespeare used the English sonnet form for his own sonnets, or at least most of them. In Sonnet 33, for example, the volta comes after the eighth line (see also Post, page 11-12). Sonnet 18, Sonnet 29, Sonnet 44 and Sonnet 94 are also "Petrarcan" in the sense that the volta comes after line 8.
"Sonnet" 126 is unusual in two ways: it entirely written in rhyming couplets and is only 12 lines long instead of 14.
The above deviations show that Shakespeare used the English sonnet form less consistently than the term "Shakespearean sonnet" suggests.
The publication of Philip Sidney's sonnet sequence Astrophil and Stella in 1591 started a fashion for sonnet sequences, such as Edmund Spenser's Amoretti (1594), Samuel Daniel's Delia (1592) and Michael Drayton's Idea's Mirror (1494). This shows that Shakespeare was definitely not the first poet who used the English sonnet form successfully.
The fashion for sonnet sequences had probably already passed by the time Shakespeare's sonnets were published in 1609. However,
The greatest of the succeeding sequences [i.e. succeeding Astrophil and Stella] was undoubtedly Shakespeare's, (...).
(Wynne-Davies, page 260)
This is probably why the term "Shakespearean sonnet" is more common than "English sonnet".
T. H. W. Crossland (The English Sonnet, page 38) makes a more radical claim (emphasis added):
This form of sonnet was
written before Shakespeare, but Shakespeare
appropriated it to himself, and every one of
his sonnets is so rhymed. Even in Sonnet 145
the rhyme scheme is maintained, and the
sonnet "prologue" to Romeo and Juliet is
similarly rhymed. The form is usually known
as the Shakespearean. We call it the English
sonnet pure and simple, because it was the
first perfect form of sonnet to take root in the
language. It is doubtful whether since the
time of Shakespeare a really satisfactory
sonnet in that form has been written. All
manner of poets have tried their hands and
their wings. Perhaps, with the single exception
of Michael Drayton, they have failed, and
Drayton may be said to have succeeded in
only one sonnet. In a sense, possibly, we may
regret that Shakespeare handled this beautiful
form with such mastery; for after him, flight
in it seems not only vain but presumptuous,
and the most self-reliant poet will think twice
before obeying an impulsion which seems
likely to result in "four quatrains clinched by
This implies that the English sonnet is often called the Shakespearean sonnet because no one has mastered the form like Shakespeare, neither before nor after him.
- Crossland, T. H. W.: The English Sonnet. London: Martin Secker, 1917.
- Post, Jonathan F. S.: Shakespeare's Sonnets: A Very Short Introduction. Oxford University Press, 2017.
- Schabert, Ina (editor): Shakespeare Handbuch. Die Zeit - Der Mensch - Das Werk - Die Nachwelt. Fifth revised edition. Kröner, 2009.
- Wynne-Davies, Marion (editor): The Renaissance. A Guide to English Renaissance Literature: 1500–1660. Bloomsbury Guides to English Literature. London: Bloomsbury, 1994 .
 This should not be taken to imply that Surrey consistently used this rhyme scheme, as Peter Shor pointed out in a comment. For example, Surrey's sonnet I never saw my Lady lay apart uses the rhyme scheme ABBA CDDC EFFE GG. It is also worth comparing Wyatt's and Surrey's translations of Petrarch's sonnet 140: Surrey's translation uses the rhyme scheme ABAB CDCD EFEF, whereas Wyatt's translation attempts to follow the rhyme scheme ABBA CDDC EFFE GG but deviates from it in the third quatrain (assuming that "die" in the last couplet rhymed with "faithfully" according to the rules of Early Modern English).
 So far, I have been unable to find out who coined the phrase "Shakespearean sonnet".