Who changed the Jews' names?

Dara Horn - whose novel, All Other Nights, I hugely enjoyed last year - blasts away the idea that Jewish immigrants to America, coming through Ellis Island, were forced by immigration officials to change their names:

Ellis Island officers never wrote down immigrants’ names. Instead, they worked from ships’ manifests, which were themselves compiled by local officials at the point of embarkation. Even overseas, passenger lists were likewise not generated simply by asking immigrants for their names. Rather, they were drawn from passports, exit visas, and other identification papers.

The reason for this was simple: Errors cost the shipping company money. A mistake on a manifest, such as a name that was not corroborated by other documentation (whether legal or fraudulent), would result in the forced deportation of the person in question back to his point of departure — at the shipping company’s expense. Of course, many Jewish immigrants’ names were changed upon coming to America. Without exception, however, they changed their names themselves.

She suggests that the immigrants wanted to integrate into American society, but were embarrased that they had shed such a defining part of their Jewish identity so easily, and so the myth arose that it had been forced on them.

It makes sense; there are numerous stories of Jewish immigrants completely re-inventing themselves on the boat on the way to America (throwing off their sheitls once the shtetl was safely behind them, etc), or shortly after, so I don't see why they wouldn't change their names themselves as well.

It is unclear to me, though, when they were supposed to have done so; presumably, in the days and months after landing, which means that the Ellis Island records would still show their "Jewish" names. If so, there must be many families who have looked for their ancestors' records in the extensive Ellis Island archives who already know that the story told to them by Bubbe and Zaide wasn't quite true....