“Liberty Enlightening the World” is the official name, but “Statue of Liberty Enlightening the World” was so frequently used that the famous statue in New York harbor is now called the “Statue of Liberty.” The names have not been trademarked.
 
“The monument is to be a colossal state of Liberty, represented as enlightening the world” was cited in September 1875. “The colossal statue of ‘Liberty Enlightening the World’” was cited in April 1876. “The Bartholdi Statue of Liberty” was the title of an article in Scientific American in 1880. The statue was dedicated on October 1886.
 
   
Wikipedia: Statue of Liberty
The Statue of Liberty (Liberty Enlightening the World; French: La Liberté éclairant le monde) is a colossal neoclassical sculpture on Liberty Island in New York Harbor in New York City, in the United States. The copper statue, designed by Frédéric Auguste Bartholdi, a French sculptor, was built by Gustave Eiffel and dedicated on October 28, 1886. It was a gift to the United States from the people of France.
 
The statue is of a robed female figure representing Libertas, the Roman goddess, who bears a torch and a tabula ansata (a tablet evoking the law) upon which is inscribed the date of the American Declaration of Independence, July 4, 1776. A broken chain lies at her feet. The statue is an icon of freedom and of the United States, and was a welcoming sight to immigrants arriving from abroad.
   
29 September 1875, Springfield (MA) Daily Republican, pg. 5, col. 3:
France.
PROPOSED FRENCH MEMORIAL MONUMENT FOR THE AMERICAN CENTENNIAL.
(...)
The monument is to be a colossal state of Liberty, represented as enlightening the world.
 
9 November 1875, Indianapolis (IN) Journal, pg. 4, col 3:
We have received from Paris a circular relatve to a unique and interesting Centennial project which has originated in that city. It is proposed by some liberty-loving Frenchmen, who desire to aid in commemorating our Centennial anniversary, to unite with patriotic Americans in erecting a colossal statue of Liberty on an island in New York harbor. To quote from the circular: “In the middle of New York harbor, on a little island belonging to the Union, facing Long Island, where the first blood has been shed for Independence, there will be be raised a colossal statue, showing its grand figure in the space horizoned by the large cities of New York, Jersey City, and Brooklyn. At the entrance of that vast continent, full of new life, where ships meet from all points of tha world, it will look as springing up from the bosom of the deep, representing Liberty enlightening the world. At night a luminous aureola projected from the head will radiate on the far-flowing waves of the ocean.”
 
Chronicling America
23 November 1875, Public Ledger (Memphis, TN), “France and America,” pg. 1, col. 2:
At this news M. barthold went and found the manufacturer and said to him: “I desire to apprise you, Monsieur, that the statue of liberty will be 27 metres in height.”
 
Chronicling America
30 April 1876, New Orleans (LA) Republican, “The Statue on Bedloe’s Island” (New York Herald), pg. 6, col. 5:
Yesterday afternoon an informal meeting was held in the office of Compagne Generale Transatlantique, by Mr. Auguste D’Ouville, of said company, and Mr. Adolph Salmon, of No. 14 Wall street, both members of the Union Franco-Americaine, at Paris, which is about placing on Bedloe’s Island, in the harbor of New York, the colossal statue of “Liberty Enlightening the World,” the cost of which is 1,800,000 francs, and the height from the base to the pedestal 105 feet.
 
14 June 1876, The Daily Graphic (New York, NY), pg. 3, col. 4:
The cast of the wrist of M. Bartholdi’s statue of “Liberty Enlightening the World,” to be set up on Bedloe’s Island, has been taken to the studio of M. Manduit, Rue de Chazelles, Paris.
 
OCLC WorldCat record
The Bartholdi Statue of Liberty
Edition/Format: Article Article
Publication: Scientific American, v43 n4 (18800724): 49
Database: CrossRef
 
OCLC WorldCat record
The Great Statue of Liberty
Edition/Format: Article Article
Publication: Scientific American, v47 n25 (18821216): 384
Database: CrossRef
 
OCLC WorldCat record
An appeal to the people of the United States in behalf of the great statue, Liberty enlightening the world.
Author: William Maxwell Evarts
Publisher: New York : [publisher not identified], 1882.
Edition/Format:   Print book : English
 
OCLC WorldCat record
Liberty enlightening the world. : Edition prepared expressly for and presented with the compliments of the Travelers Insurance Co. Hartford, Conn. : This engraving represents the colossal statue by Bartholdi. Presented by the French people to America : as it will appear on its pedestal on Bedloes Island in New York Harbor. : Height of pedestal above high water 177 ft. 9 in. Height of statue 151 ft. 2 in. Total height from water 328 ft. 11in.
Author: William Maxwell Evarts; Joseph W Drexel; James G Batterson; Frédéric Auguste Bartholdi; Richard Morris Hunt; All authors
Publisher: [New York] : Copyright 1883 and published by Root & Tinker, Tribune Building, New York., [1883]
Edition/Format:   Image : Graphic : Original artwork : English
Database: WorldCat
Summary:
Subject: Image of the Statue of Liberty by Bartholdi, standing on its pedestal designed by Richard Morris Hunt, with many people walking along the base of the statue and around the pedestal. Behind the statue, ships and sailboats are in the harbor and a cityscape of New York City can be seen.
 
Google Books
Model of the Statue of “Liberty Enlightening the World,”: Issued by the American Committee, in Aid of the Pedestal Fund
Committee (33 Mercer St., New York), 1884
 
Google Books
Oration at the Unveiling of the Bartholdi Statue of Liberty Enlightening the World, October 28, 1886
Chauncey Mitchell Depew
1886
 
Google Books
The Statue of Liberty Enlightening the World,
Described by the Sculptor,
Frederic Auguste Bartholdi
New York, NY: The North American Review
1890