Does Clink Street take its name from the prison or vice-versa?

In Southwark, between the Cathedral and the Globe Theatre runs Clink Street. It is in that dark and dingy alley that such that remains of the notorious prison, which is now rebuilt as a museum, is found.

Narrow, dark, and cobbled, it is well-known for the chase-scene in the David Lean film Oliver Twist. http://en.wikipedia.org/wiki/Clink_Street

Now the word 'clink' is widely used in Britain for prison. 'He did a year in clink', 'At this rate he will end up in clink' etc.

What I am unsure about, is whether 'clink' derives from onomatopoeia, the sound of metal doors closing, keys, chains and fetters etc. and in that way became an everyday term for prison. Does Clink Street take its name from the prison, or did prison become known as 'clink' because a notorious one (from the 12th to 18th centuries) was located in Clink Street.

It is not clear from the quotations in the OED.

The Eric Partridge–edited third edition of Francis Grose, A Classical Dictionary of the Vulgar Tongue (1796, originally), has an interesting entry by Grose [and followup note by Partridge]:

CLINK. A place in the Borough of Southwark, formerly privileged from arrests; and inhabited by lawless vagabonds of every denomination, called, from the place of their residence, clinkers. Also, a gaol, from the clinking of the prisoners' chains or fetters: he is gone to clink. [The place was in Grose's day, as it had long been, the noted gaol: The Clink. Originally a sanctuary district: 'the liberty of the Clink' was, then, the sanctuary afforded by that district, By a play on liberty, clink came to mean the Clink gaol; thence the Clink became any gaol. ... While clink is probably connected with clinch, to clutch, etc., whence clinch also = a gaol, it is probable that the onomatopoeic associations may have helped to popularize clink as gaol.]

Grose, at least, seems to think that clink in the sense of "gaol" comes from the sound of clinking chains and not from the place in Southwark "formerly privileged from arrests."

For its part, Merriam-Webster, Webster's Word Histories (1989), seems fairly confident that clink in the sense of "prison" derives from the Southwark Clink:

clink ... [fr. Clink, a prison in Southwark, borough of London, England, prob. fr. Clink, a part of the Manor of Southwark]

Followup (August 30, 2021)

Francis Grose, A Provincial Glossary; with a Collection of Local Proverbs, and Popular Superstitions (1824) reports that "the Clink" was also known as "the Mint":

A clinker. An inhabitant of the Mint or Clink, formerly a place privileged from arrests, the receptacle of knaves and sharpers of all sorts.

In contrast to Grose, who attributes "the Clink" to the clinking sound of chains, Maurice Jonas, Shakespeare and the Stage With a Complete List of Theatrical Terms Used by Shakespeare in His Plays and Poems (1918) says the the clink derives from a different sense of the verb clink:

The Clink was the name of the noted prison in Southwark; the name is derived from the word "clink," to fasten securely.

John Hines, Nathalie Cohen & Simon Roffey, "Iohannes Gower, Armiger, Poeta: Records of His Life and Death," in A Companion to Gower (2004) indicates that "the Clink" may have been in use as the name for the prison by the beginning of the fifteenth century:

Southwark also had five prisons by the late Middle Ages: the Marshalsea, the King's Bench, the Counter, the White Lion (Surrey County Prison) and the Clink.

The Clink was not originally a major institution, according to Edward Brayley, A Topographical History of Surrey, Volume 5 (1841):

Clink Liberty—There was a small place of confinement on the Bankside called the Clink from being the prison of the "Clink Liberty or manor of Southward'" belonging to the bishops of Winchester. It was used for the confinement of disorderly persons and other petty offenders. In old time, Stow says, it was a prison "for such as should brabble, fray, or break the peace of said Banke, or in the Brothell houses or Stews." About 1745, the old prison, which stood at the corner of Maid-lane, being in great decay, was abandoned, and a dwelling on the Bankside was appropriated in its stead. The latter was burnt during the riots in 1780; and no other prison for this liberty has since established.

"Stow" is John Stow, The survey of London Containing the Original, Increase, Modern Estate and Government of That City, Methodically Set Down (1598/1633), whose original wording is as follows:

There be also five Prisons, or Gaoles [in Southwarke].

The Clinke, on the Banke.

...

Then next [to "this row of Stewes"] is the Clinke, a Gaole or Prison for the Trespassers in those parts; namely, in old time for such as should brabble, fray, or breake the peace on the said Banke, or in the Brothell houses, they were by the inhabitants thereabout apprehended and committed to this Gaole, where they were straightly imprisoned.

Reference to "the Clinke" appear in A.F., Certeine Comfortable Expositions of the Constant Martyr of Christ, M. Iohn Hooper (1580) and in "The Examination of me Henry Barrowe, the Nineteenth of Nouember, 1586, Before the Arch Bishope, Arch Deacon, and Dr. Cuffins, as Neere as My Memorie Could Carry, Being at Lambeth," reprinted in The Harleian Miscellany (1745).

The earliest mention of the Clinke that an Early English Books Online search turns up, however, is from "The effect and summe of the last examinacion of that faithfull instrument of God Iohn Bradforde in the Churche of Saincte Marie Oueries the 29. daie of Ianuarii .1555", All the Examinacions of the Constante Martir of God M. Iohn Bradforde Before the Lorde Chauncellour, B. of Winchester the B. of London, [and] other co[m]missioners (1561):

After the excommunicacion was red, he was deliuered to the sheriffes of London, and so had to the clinke, from thence to the coūter in the pultry, where he remaineth close, without al company, bokes, paper, penne or ynke, loking for the dissolucion of his bodye, in ye which god grant to hym hys sweete mercye: throughe Christe oure Lorde. Amen.

The Church of St. Mary Overie is in Southwark, so it would be natural for a prisoner being examined for heresy at that Church by ecclesiastical authorities to be held in the Clink nearby, especially as the Clink was maintained (originally) by the Bishop of Winchester. John Foxe's Martyrology mentions "the Clinke" on multiple occasions in connection with imprisonments that occurred in 1556 and 1557 during Queen Mary's reign.

With regard to whether the street was named for the prison or the prison for the street, John Stow, A Survey of the Cities of London and Westminster (1753) provides what appears to be a complete list of the streets that fall wholly or in part in the Clink Liberty:

In the Clinke-Liberty:

Part of Church-way ; Part of Dirty-lane, Rochester-yard, Winchester-street, Primrose-alley, St. Saviour's-dock-head, Winchesteer yard, Stoney-street, Clinke-street, Clinke-yard ; Part of Deadman's-place, Globe-alley, Naked-boy-alley, Vine-street, Maid-lane, Marshal-street, Fountain-alley, Horse-shoe-alley, Rose-alley, Bear-garden, New Thames-street, Morse's-alley, White-hind-alley, Barton-court, Gardeners-lane, Back-side, Willy-street, Red-cross-street, Angel-court, Casttle-street, Castle-lane ; Part of Fishmongers-alley, Red-cross-alley, Red-cross-court, Queen-street, Worcester-street, White-crest-street, Little-hndy-leg-walk, Great-handy-leg-walk, Lower-street, Ewer's-street, Duke-street, Prince's-street, Low-man's-pond, Orange-street, and Pepper-street.

And Samuel Lee, A Collection of the Names of the Merchants Living in and About the City of London (1677) mentions two merchants who were residing or doing business in Clink Street:

James Smith Clink street

...

Daniel Van Pray Clink street Mary-gold stairs Southw[ark].

Evidently the name Clink street was in use by 1677, but I haven't been able to find any older references to a street of that name. The earliest mention of "Clink street" that I've been able to confirm is more than 120 years younger than the earliest confirmed mention of "the Clinke" [prison].

Interestingly, "clinke" seems to be used in a generic sense as early as Anne Dowriche, The French Historie, That Is, A Lamentable Discourse of Three of the Chiefe, and Most Famous Bloodie Broiles that Haue Happened in France for the Gospell of Iesus Christ (1589), which contains these two instances in describing prisoners during the religious wars in France of 1557–1572:

And that which was the worst, in prison where they were, / The theeues and bloodie murtherers did find more fauor ther. / For they that death deserude were taken from their clinke, / And in their colde & vglie pits which breathd a deadly stinke / These men were thrust & bound, & kept with watch & ward, / That al accesse of worldly ioy from them might quite be bard.

...

The godlie trapped thus, and thus to thraldome sent; / As sheepe vnto the slaughter they to prison meeklie went. / Vvhere lying in the Clinke their feete and hands were bound, / And by the cruell Iailors were laid prostrate on the ground. / Then Mandelot [governor of Lyons] commands the hangman for to call, / Whom he enioynes to enter in with axe to kill them all.

In the late 1500s, the association of the Clinke in Southwark with religious persecution very likely informed its application (by Dowriche) to other prisons with a similar orientation.

Meanwhile, an explicit connection between clinke and the sound of fetters or shackles appears at least as early as The Trimming of Thomas Nashe Gentleman, by the High-Tituled Patron Don Richardo de Medico Campo, Barber Chirurgion to Trinitie Colledge in Cambridge (1597):

... so say I to thee, Nashe come forth, be not ashamed of thy selfe, stretch out thy legs that euery step thou goest, thy shackles crying clinke, may remember & put thee in minde of all thy goodnes and vertue: I am glad to see thee in this prosperitie, thou neuer wert so rich as now, thou neuer hadst so much money as would buy so faire a payre of fetters: in very deed thou art beholding to thy keeper that will trust thee with so faire a payre of fetters, neither would he if hee had thee not by the legge: ...

But whether this connection existed at the time of the naming of the Clinke prison, I do not know.