Local customs and legal definitions of naming areas of territory and defining municipal boundaries vary WIDELY from place to place around the U.S.
For example, I grew up in New England, where there is no unincorporated land. (OK, not NONE, but in New Hampshire, for example, where I grew up there was perhaps a couple hundred square miles in the whole state that was officially unincorporated). Everything was either a “city” or a “town” which depended solely upon the type of government of the locality, not the population. Cities have mayoral forms of government. Towns have “council” forms of government, usually in the form of a “board of selectmen.” I grew up in the Town of Hudson, NH; which has a population of about 30,000 people. For contrast, the City of Claremont has a population of about 10,000 people. Both towns and cities are considered “incorporated” places, and there is darned near nowhere in New England (other than perhaps northeastern Maine) that isn’t part of a town or a city.
Many large cities around the nation have absorbed neighboring towns and cities, and these areas vary widely with regard to level of local autonomy and local naming practice. New York City, for example, absorbed Brooklyn, Astoria, Jamaica(both part of the borrough of Queens), The Bronx, Staten Island, etc. Georgetown was absorbed by what later became Washington D.C. Roxbury, Charlestown, Dorchester, Allston, and Brighton all used to be seperate towns around Boston; they are now all considered part of Boston legally, even if they have retained some of their local character. Some neighborhoods grow up locally inside of greater cities, like Lakeview or Hyde Park in Chicago, and some independent cities exist inside of metro areas that makes them indistinguisbible from the larger city; Toronto and Los Angeles have areas like this.
Mailing address gives little indication as to political or cultural boundaries that really exist. “New York, NY” will only deliver to addresses on Manhatten; Brooklyn, NY for example, is a separate mailing address. Northern New Castle County Delaware has the opposite extreme; Officially there are only 5 mailing addresses: Wilmington, Newark, Hockessin, Bear, and New Castle; even though there are DOZENS more “towns” in these areas.
Ultimately, where am I heading with this? There is no “official” system as to what to call these areas. Neigborhood is a safe term; if you say “I come from the River Park neighborhood of South Bend” it sounds fairly normal. Unless people from River Park use a different term locally like “village” or “borrough” or “community” or “quarter.”