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:

From Wikipedia:

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.

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.

From Wikipedia:

• An article is a word that is used alongside a noun (…) to specify grammatical definiteness of the noun, and in some languages extending to volume or numerical scope. The articles in English grammar are the and a/an, and in certain contexts some. "An" and "a" are modern forms of the Old English "an", which in Anglian dialects was the number "one" (compare "on" in Saxon dialects) and survived into Modern Scots as the number "owan". Both "on" (respelled "one" by the Norman language) and "an" survived into Modern English, with "one" used as the number and "an" ("a", before nouns that begin with a consonant sound) as an indefinite article. […]

• The definite article is used to refer to a particular member of a group or class. It may be something that the speaker has already mentioned or it may be something uniquely specified.

• However, recent developments show that definite articles are morphological elements linked to certain noun types due to lexicalization. Under this point of view, definiteness does not play a role in the selection of a definite article more than the lexical entry attached to the article.

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.