My guess is that Wikipedia is wrong, which is also borne out by their entry for the city of Corning. That identifies the city as the western part of the town.
In New York (also New Jersey, and I don't know how many other states do things this way), there are often geographical divisions within counties that include several smaller areas, usually identified as either cities or villages or just plain not identified as anything other than a name on a map. These divisions, in New York, are called towns and, in New Jersey, they are called townships. Often the town or township includes an area with the same name as the town, which is often the hub (either population or business) of the town or township. Please note that each of these places has its own zip code. When sending mail to someone who lives in one of these places, the correct way to write the bottom line is Northport, New York, 11768 (for example). The fact that it is technically part of the town of Huntington is never mentioned and people who live there don't tend to think of themselves as living in the town of Huntington, although they do think of themselves as living in Suffolk County.
Although I don't have actual knowledge, I believe it is most likely that the city of Corning, New York is contained within (and is part of) the town of Corning, New York. When naming places in New York that are included in towns, I tend to ignore the town and just enter:
city, county, state, country
as the location.
A prime example of this is the Town of Huntington, which is on Long Island in New York. Long Island, itself is a sort of non-place, as location names go. It is comprised of two counties - Nassau and Suffolk. Within each county are several towns. Suffolk includes the town of Huntington, which includes (communities/villages/cities/whatever that are the lowest level place name) Huntington, Centerport, Northport, East Northport, Half Hollow Hills, Commack, Eaton's Neck, Lloyd's Neck, Lloyd Harbor, and probably a couple of others I can't think of at the moment.
I'm not sure what categorization wants to do about this kind of thing - whether to have structure of:
state -> county -> town -> low-level place name
or to eliminate town name from category structure. That's a question for them to decide, once they understand how the system works in practice.
