Child #286, Roud Folk Song Index #122.

Those two ships have had an awful lot of names over the years; sometimes, too (as here) the other ship -the Turkish Galley, Turkey Silveree etc.- gives the song its name. In a broadside example of the late 17th century (given by Child as his version A) the ship is The Sweet Trinity, built by Sir Walter Rawleigh, and the enemy is simply "a false gallaly", but The Golden Vanity is the best-known name; the enemy being variously French, Spanish, Turkish and so on. In fact, the name seems to have been mutable for a long time; Child also mentions The Golden Victorie, The Gold Pinnatree and The Golden Trinitie. Other examples include:

The Bold Trellitee (Catskills)
The Gallant Victory (Massachusetts)
The Golden Furnity (Utah)
The Golden Merrilee (Florida)
The Golden Vallady (Nova Scotia)

Often the song has been given the Golden Vanity title by collectors for convenience, where their source actually had a quite different name for the ship, such as:

The Golden Silveree (North Carolina)
The Golden Willow Tree
The Green Willow Tree (Kentucky)
The Mary Golden Tree
The Silver Family
The Bold Trinitee (Virginia)
The Sweet Kumadie (Aberdeenshire)

There are five traditional sets from the Ozarks, recorded in the 1960s, at The Max Hunter Folk Song Collection:

Hunter #61 Turkish Sugarlee
Hunter #393 Turkish Revillie
Hunter #519 Green Willow Tree
Hunter #595 Little Ship
Hunter #747 Golden Willow Tree

As might be expected, there are a good few sets in the DT, mostly from the USA:

THE SWEET KUMADEE Scottish set; traditional source not noted.
THE LOWLANDS LOW (7) 1931 set taken from Randolph's Ozark Folk Songs. The names of the ships are reversed; Turkish (Turkey) Shivaree and Green Willow Tree.
THE LOWDOWN LONESOME LOW Taken from John A and Alan Lomax, Our Singing Country. Includes a Turkish Reveille.
THE GREEN WILLOW TREE 1951 set from McNeil's Southern Folk Ballads. (Turkey Shevelee).
THE GOLDEN VANITY (6) Source unknown, perhaps heard from Pete Seeger. Spanish enemy.
THE GOLDEN VANITY 1950s set taken from Helen Creighton's Folksongs from Southern New Brunswick. Turkish enemy.
THE BOLD TRELLITEE Taken from Cazden's Folk Songs of the Catskills. Turkey Degree.
GOLDEN VANITY No source named. Turkish Revelry.
LOWLANDS (minstrel show) Brief parody.
SINKING OF THE GRAF SPEE Parody.

There's also some material in the Forum, particularly:

THE GOLDEN VANITY Set from The Penguin Book of English Folk Songs, noted by Anne Gilchrist from W. Bolton, Southport, Lancashire, in 1906. Spanish enemy.

Entry at The Traditional Ballad Index:

http://www.csufresno.edu/folklore/ballads/C286.html

A broadside copy at Bodleian Library Broadside Ballads:

The Golden vanity, or The low lands low Printed between 1849 and 1862 by H. Such, Newsvender, &c. 123, Union Street, Borough, London. (Spanish Galleon.)

A rather long-winded way of answering your questions, but it's more interesting than saying (a) usually (b) probably, but well before it acquired the Turkish Reverie title and (c) nobody knows for sure!