As I recently tried to research one arm of my family I think that most posters on here are greatly overestimating both how standardized spelling was back in the day and also ignoring how complex other factors are.
My family is from Finland, which was under Swedish control for centuries and the records from that time are from the church and the records were kept in swedish by individuals who worked in them.
This means although my Great-great grandfather was named Lauri all of the records from his homeland were under Lars and while he didn’t pass through Ellis Island his wife did a few years later and they have him down as Luis.
While we like to think of these early immigration inspections stations like Ellis Island as embracing the masses the federal government took over immigration in 1890 mostly due to concerns that Europeans that were not from the Anglo-Saxon branches would destroy American culture after steam-powered ships lowered the cost and more Italians, Greek, Slavic, Ugrian, and other peoples started coming over.
Only the steerage (poor) passengers had to go to Ellis Island for further processing as the more financially well off passengers were allowed to just leave from the docks in Manhattan where their ships actually arrived to. Remember that this is an era when eugenics and the desire for a “superior stock” was the norm among officials but the denial rate was also very low. Also there was no requirement for registration once you were in the country as an alien until well into the 1900’s if you weren’t from Asia.
The point being that Ellis Island was less about providing official papers until ~1920’s and later and more about filtering out “defective people” and even if it had been misspelled at ellis island it wouldn’t have impacted how someone spelled it when they got to their final destination.
But due to non-standardized spellings, and actually the fact that our family naming scheme is fairly recent mixed in with the fact that those who did pass through the immigration stations were poor and likely illiterate there are a dozen reasons the spellings could have changed.
Add in the language barrier and the fact that some inspectors were more interested in seeing if they could extract a bribe and the accuracy of the records is pretty amazing.
Some people came from areas where surnames were not common yet or they may not have liked the ones from the old country or possibly were assigned new names by immigration personnel who could not understand or spell the names given.
But without direct handed down information it would be hard to tell.
In my case on the Paternal line it was a mix with my g-g grandfather keeping his Finnish first name and my g-g grandmother using the English form and they used the Swedish form of their last name but using the Northern Swedish spelling. To complicate this census records also mix this up, often with state and county officials reverting to the Swedish church versions but sometime using English versions. As I think that my relatives remained functionally illiterate through the remainder of their lives I doubt that they noticed but it did become more consistent as their children aged and probably helped them.
The point is that any definitive answer on systemic differences is probably lost to time and any argument that claims differently would require specific proof. There was a lot of bribery and corruption in the system though so I guess that is a pretty good reason to believe that that other portions of the process were probably less than rigorous too.