I'd say you are correct about the placement of the generic word being the reason for using (or not using) 'the' , and all your examples are phrased correctly. Notice that 'Falls' is different also in that it is a plural.
Edit: I have found the answer:
In English, nouns must in most cases be preceded by an article that specifies the presence or absence of definiteness of the noun. The definite article is the in all cases other than generic references, which use the zero article (i.e., the absence of an article), while indefiniteness is expressed with a or an for singular nouns or the zero article for plural or non-count nouns.
From Monmouth University:
The definite article 'the' is used before both singular and plural nouns when the noun is specific. The names of geographic places are specific names and may require definite articles:
names of rivers, oceans, seas, geographical areas, deserts, forests, gulfs, peninsulas, groups of lakes (the Great Lakes), mountain ranges, and chains of islands.
No article is necessary before the following specific nouns:
Singular names of countries or territories, cities, towns, states, streets, lakes, bays, mountains, continents, islands, languages, sports, academic subjects.