I think if you're on the New York page and you write
You want the current page to be listed under America, not by its real name New York, but by its alias Big Apple.
So the thing after the pipe is an alias for the current page, not an alias for the category.
So, go to an Ellis Island immigrant and write
[[Category:Immigration Arrival Ports - America - New York]]
No alias, because the immigrant keeps his own name.
Click on the red link and write
[[Category:Immigration - Ports of Entry - America|New York]]
and so on. Assuming the names are OK - no idea what the rules are.
PS I now see that one's already done, but using the long names without the short aliases
PPS That's not right is it, The alias doesn't change the name that appears, it only changes how it sorts. Bummer.