When I use the API, both “Cortland NY” and “Cortland County NY” return the same coords.
42.5842136, -76.0704906 would be correct for Cortland County.
42.60026 -76.17773 would be correct for The City of Cortland, which your second link shows.
I guess my issue is I cannot get the API to return “The City of Cortland”. Both Cortland and Cortland County return the same coords with a different name for me.