Entry 2 for "Turtle" via Dictionary.com:
1. a turtledove.
before 1000; Middle English, Old English < Latin turtur (imitative)
Expanded word origin of "turtle" via Dictionary.com:
"turtledove," Old English turtle, dissimilation of Latin turtur "turtledove," a reduplicated form imitative of the bird's call. Graceful, harmonious and affectionate to its mate, hence a term of endearment in Middle English. Turtledove is attested from c.1300.
From the Morrish Bible Dictionary:
Where the word 'turtle' occurs in the A.V. [KJV] the 'turtle-dove' is always to be understood.
From Easton's Bible Dictionary:
The Latin name of this bird, Turtur, Is derived from its note, and is a repetition of the Hebrew name Tor.
From Smith's Bible Dictionary:
Turtur auritus (Heb. tor). The name is phonetic, evidently derived from the plaintive cooing of the bird.
From Fausset's Bible Dictionary:
tor; Latin, tur-tur , from imitation of its cooing note.
So yes, "turtle" was used for the bird also known as a "turtledove" in the 1600s. The two words were used interchangeably at the time for the same bird. This was the case even before the King James Version, as can be seen in the examples below (note that several of the older Bible versions have chapter but no verse numbers).
King James Version, 1611
Geneva Bible, 1587
Bishop's Bible, 1568
Taverner's Bible, 1551
Matthew Bible (2nd edition), 1549
Great Bible, 1541
Matthew Bible, 1537
Coverdale Bible, 1535
Notice that the Coverdale Bible does use "turtill doue" (turtle dove) in what would later be verse 6 and verse 8, whereas all the later versions just use turtle in verse 8. The two terms were used interchangeably.
A modern-day equivalent would perhaps be the terms "firefly" and "lightning bug." They are two terms that are used interchangeably for the same insect, which is in fact a beetle. The names "firefly" and "lightning bug" are derived from the way the insect is able to "fly around and light itself up like a flash of fire," or that it is a "bug that is able to give a flash of light like lightning." While the two terms are used interchangeably today, it is possible that they would not be used interchangeably 400 years from now.
Additionally, according to Dictionary.com and Merriam-Webster, "turtle" as in "tortoise" was not really used until shortly after the King James Version was published.
Word origin of "turtle" as in "tortoise" via Dictionary.com:
1625-35; alteration (influenced by turtle2 [turtledove]) of French tortue < Medieval Latin tortūca tortoise
Expanded word origin of "turtle" as in "tortoise" via Dictionary.com:
reptile, c.1600, "marine tortoise," from French tortue "turtle, tortoise," of unknown origin. The English word is perhaps a sailors' mauling of the French one, influenced by the similar sounding turtle (n.2). Later extended to land tortoises; sea-turtle is attested from 1610s.
Word origin of "turtle" as in "tortoise" via Merriam-Webster:
modification of French tortue, from Late Latin (bestia) tartarucha, feminine of tartaruchus of Tartarus, from Greek tartarouchos, from Tartaros Tartarus; from Mithraic and early Christian association of the turtle with infernal forces
First Known Use: 1612