A four month old Stephen, accompanied by his mother, elder brother, and two half-sisters, arrived in New York aboard the S.S. Gellert, on August 11, 1875. Meaning that he was born in 1875, likely on March 17th, as March 17th was the only consistent feature in his varying dates of birth, plus it's corroborated by his aged being stated as "4" months on the Passenger List of the ship which brought him to the United States.
I've not been able to irrefutably prove precisely when (meaning on what ship, what day, etc.) his father arrived in the United States. As he travelled alone, it's far more difficult to pin down. But the following sources all claim that he arrived in the United States during 1873, which explains why they found it necessary to pretend Stephen was three years older than he actually was.