Glenn's Falls is not a park

Some places have City in the name of the place, like New York City, Oregon City, Rapids City. Sometimes the place really doesn’t seem large enough to merit the name, like Siler City, NC, although I suppose in the 19th century it was more prominent in its part of the state.

every city is "city of" or "x city"

I think a hasty generalization has led to a false assumption. Many cities are incorporated as "City of x" or "x City". But, "many" is not equivalent to "all".

Who knows. Would NYC even be considered one "city", rather than a set of burroughs, for this purpose? Or are the names perhaps normalized by population? If so, would they remove outliers as many statistical packages will do for you, but in this case they should leave them in, so if they were removed automatically that would not be great. Or if they just went with one name = one count then is there a minimum cutoff? Or a nearness criteria e.g. places near NYC still get swept up into it? And like how much wood would a woodchuck chuck if a woodchuck could chuck wood? So many unanswered questions here...