It seems certain that in 1907 Eugenia Farrar did make a broadcast over Lee de Forest's experimental Parker Building station, most likely that October. However, over time the story has gotten somewhat embellished. In particular, the supposed contemporary New York Herald report apparently does not really exist, which makes it difficult to ascertain what actually happened.

The American Swedish Monthly, January 1955, pages 10, 26:

I  Was  First  to  Sing  Over  the  Radio

By  Eugenia  H.  Farrar
I was born in Stockholm in 1875. During my 80 years there have been many thrilling experiences, many adventures. The life of a professional singer never is a quiet one. Yet I believe that my greatest thrill of all was on that October day in 1907 when I sang for Dr. Lee de Forest over his wireless telephone, as it then was known--the first radio performance, I was told.
    Good looks and a voice go well together, especially if one is to make a living by singing. I like to think I inherited these traits from my parents. My mother, who had been Fredericka Wilhelmina Berglund before she married, was a remarkably beautiful woman. My father, John M. von Boos, a graduate of Uppsala University, had a marvelous basso voice and could have won fame as a professional singer had he not chosen to be an agricultural expert. His voice actually led to my singing. In summer when we were on the farm, not far from Stockholm, father would take me driving into the deep woods, tie the horses and we would walk further. Meanwhile he would sing at the top of his voice and I would sing with him. When I was 13, I became a pupil of Professor Julius Gunther, director of the Royal Academy of Music. Fortunate it was that I made the most of my opportunities, for I was barely 17 years old when my father died and our reduced circumstances impelled my mother to bring me to the United States which, she believed, and rightly, offered an excellent chance for my career. We lived in New York, and life was comfortable and benign. I was singing under my married name of Eugenia Farrar when Dr. de Forest invited me to join him in an experiment with something I knew absolutely nothing about, but which he had named his wireless telephone. Early de Forest wireless telephone equipment
    For those too young to remember events almost 50 years ago, Lee de Forest, who today lives happily in Los Angeles, was one of the greatest inventors in radio. After Marconi's invention of the wireless, the young de Forest commenced inventing and developing with all the youthful and terrific imagination which had brought him out into the world from his birthplace in Council Bluffs, Iowa. His audion tube permitted the human voice to be carried by wireless. By 1907 when Lee de Forest said he could do a thing people believed him. Thus it was that the United States Navy had commissioned him to install his new voice radio on all the 44 vessels which Admiral "Fighting Bob" Evans was about to take on a notable cruise around the world. The first set had been placed on the Admiral's flagship in Brooklyn's Navy Yard, but there was no actual proof that it would receive or send the human voice. That was when Dr. de Forest invited me to sing. It was a whimsical and wholly humorous thing to do, considering the result. Dr. de Forest told me long after work that he wanted to have a woman singing so the Navy people would be convinced the voice was not coming from somebody planted nearby.
    Accompanied by a woman reporter, I climbed the creaking stairs to the de Forest laboratory, then on the roof of the old Parker Building at 19th Street and Fourth Avenue, New York. Coils, generators and scattered wire littered the place. In the center of all this was de Forest's wireless telephone transmitter which, I was told, utilized a crudely modulated arc generator. It was a fearsome device to me, throwing off blue sparks and an acrid stench of ozone. A horn like that of an early phonograph stood on a table, and I was told that this was the microphone into which I was to sing. In fact, I had not come to the laboratory to sing. The reporter had brought me to look at what was going on. But when Dr. de Forest learned that I was a singer he was seized with the idea, and asked me to sing over his wireless telephone.
    I sang two favorites by Carrie Jacobs Bond--I Love You Truly and Just A'Wearying For You. Frank Butler, Dr. de Forest's assistant, was at the controls in the laboratory--the first radio announcer. Only on this occasion he did no announcing. Seven miles away in the Brooklyn Navy Yard a wireless operator was copying code when the dots and dashes ceased. For a instant the ether crackle, then he heard a woman singing. Strange, he thought, a woman singing in the Navy Yard. He lifted the headphones from his ears. The singing stopped. Must be coming over the air, he thought, and replaced the headphones. Now he knew. But he was perplexed. Few people really believed that the human voice would ever come over the air.
    "Angels," he told himself. "Angels singing in the air." He ran to notify the communications officer on duty. The young lieutenant was smiling tolerantly as he entered the wireless room and adjusted the headphones. "It's impossible," he gasped.
    Later he phoned the New York Herald and reported what to him was an amazing phenomenon. The Herald next morning carried a short item about angels singing in the air. Dr. de Forest read it on his way to the laboratory and phoned the Herald to explain everything.
    Well that is the story. On several occasions I have been a guest at great dinners to Dr. de Forest, one at the New York World's Fair in 1939 when he was presented as the "father of radio" and I was asked to describe that first broadcast. There was another dinner to him in 1951 when former President Herbert Hoover spoke humorously and lovingly about the de Forest gifts to the world, but asserting that world would not be fully satisfied until de Forest had rectified one great fault with his inventions. He must make one more improvement in the radio he fathered--develop a little button to cut out the commercials.          End
Father of Radio by Lee de Forest, 1950, pages 232-233. The quoted New York Herald article, "Wireless Telephones For War Ship Fleet to be Installed at Once", actually appeared on page 2 of the Saturday September 7, 1907 edition:
    The radiotelephone work which we had been doing during the spring and summer, especially that at Put in Bay, had been closely watched by the U.S. naval officers in Washington, notably Admiral Evans' staff. The following item appeared in the New York Herald of Saturday, September 8, 1907:
    Wireless telephones are to be installed on the battleships Connecticut and Virginia of the North Atlantic Fleet, and before the warship armada sails for the Pacific the entire fleet will be equipped.
    This is attracting great attention, and is a departure almost as important as installation of wireless telegraphy on American ships. It is expected that the captains of the Fleet during their voyage will be able to converse with one another as readily at a distance of between five and ten miles as [officers on shore duty talk] from one bureau of the Navy Department to another.
    The equipment was put on board the battleships in the Brooklyn Navy Yard, and final tests were made on the eve of the ships' departure from New England waters. It was on that occasion that the first human voice actually sang into the radiotelephone transmitter. A handsome contralto singer by the name of Van Boos was invited to my laboratory to sing. The song she selected for this occasion was, I Love You Truly. It was heard by operators Smith and Wallace in the Brooklyn Navy Yard.
Radio Guide, August 18, 1939, page 12 (original scan):
Article's title display: 'And an Angel Sang!'
How  was  a  wireless  operator  to  know  he  had  heard  radio's  first  broadcast--for  it  was  only  1907!

By  Dick  Dorrance
THROUGH the quiet of the Brooklyn Navy Yard wireless room came the staccato chatter of code. With practised fingers the operator copied it down, until finally the dots and dashes halted. He glanced at the clock, noted the time on the message.
    In his headphones the ether crackled idly. Perhaps a minute passed, and then--from far away--he heard a voice singing. That was strange...a woman singing in the Navy Yard. The wireless man lifted the headphones from his ears. The singing vanished.
    Then it must be coming over the air, he decided, and listened carefully once more. The voice was really in his headphones--of that he now felt certain. The operator became just a wee bit frightened. Because after all everyone knew that voices couldn't come over the wireless--even though a few visionaries had been experimenting with such a notion.
    "Angels!" he told himself, face paling..."angels singing in the air!" Then he ran to notify the communications officer on duty. When the young lieutenant entered the wireless room there was a smile on his lips. He clamped one headphone to his ear--and the smile faded. "It's impossible," he gasped.
    In 1907 almost everyone thought such a thing was impossible. But these two Navy men, though they failed to realize it at the time, were hearing the first true entertainment broadcast ever made in the history of radio. That "angel singing in the air" was the forerunner of millions of hours of programs broadcast around the world every year.
    The singer they heard was not celestial. She was a young concert artist of the day named Eugenia Farrar. When she sang belore the crude apparatus of an equally young scientist, Lee de Forest, it was all in the nature of a lark. The young lady could hardly be expected to peer ahead a brief quarter-century and foresee the incredible industry which now brings us music, voices, entertainment and education wherever we go.
    Madame Eugenia Farrar is still alive today. She resides quietly in New York City, scene of her former vocal triumphs, and sometimes recalls the eventful day early in 1907 when she became the first person in the world to sing over the new marvel of wireless telephone.
    It was a spring afternoon when the young Swedish singer, accompanied by a woman journalist, climbed the creaking stairs to the roof of the old Parker Building, standing at the corner of Nineteenth Street and Park Avenue in New York. They found the laboratory of Lee de Forest a remarkable place.
    Coils and generators and scattered wire raced around the room. In the central position was his pioneer telephone transmitter, which utilized a crudely modulated arc generator. In operation, it proved a fearsome device, emitting blue sparks and the acrid stench of ozone. Upon the table stood a horn--almost like a regular gramophone horn of the day--which served as his primitive microphone.
    The girls evinced polite interest in all that Dr. De Forest showed them. But when he learned of Eugenia Farrar's flawless mezzo-soprano voice, he was quick in his invitation for her to sing. She stepped to the horn with a pleased little laugh and--while the transmitter crackled and buzzed--sang "I Love You Truly," followed by "Just A-Wearyin' for You." She had no accompaniment--just her own perfect voice.
    From the building in downtown Manhattan the clear notes fled across the rooftops of New York, straight to the headphones of the puzzled operator at the Brooklyn Navy Yard, seven miles away.
    The officer in charge at the wireless room telephoned the New York Herald and reported the amazing phenomenon. A short item appeared in the following morning's paper. Dr. De Forest read it while traveling downtown on the subway and, as soon as he reached his laboratory. called up the Herald to explain the origin of the "angels singing in the air."
    Today Madame Farrar looks upon her 1907 adventure with a smile. But the records show that although originally radio telephones were used by the Navy as an emergency device and considered a novelty, after this 1907 experiment radio was adopted as a vital element in fleet operation. When the U. S. fleet left the Navy Yard on a round-the-world cruise late in 1907, the De Forest wireless telephones were on every vessel--due largely to the interest of Rear Admiral Robley Evans in the feat of "angels singing in the air."
    "No," says Madame Farrar, "when I sang that day I hadn't the slightest notion what lay ahead for radio. I don't think Dr. De Forest did either. Its so different today--we thought of it then as a curiosity, an interesting novelty without any definite use..."
    Eugenia Farrar made a guest appearance on one of the major networks last fall. It was her first microphone venture since 1907.