Before US naturalization reform in the early 20th century, any court of record could record a declaration of intention to become a United States citizen or naturalize an individual. The court that recorded the declaration of intent may not have been the same as the court that finally approved the naturalization. The declaration was a preliminary step in the naturalization process and not all who declared finally naturalized although many did. You should always look for a declaration of intention to become a citizen as it may provide information beyond what is on the naturalization–or it may not. The amount of detail can vary from one location to another and from one court to another, especially before the process was standardized in the early 20th century.
But don’t limit your search for a declaration to become a citizen to the location where the naturalization was approved. People moved. They may have filed their declaration of intention in the location where they initially settled and completed the naturalization process elsewhere. Remember also that before the early 20th century reform of naturalization laws any court of record could naturalize and that most of those courts were local county courts.