The details you mentioned are Interesting, but the miss the main point.

genitive - relating to or denoting a case of nouns and pronouns (and words in grammatical agreement with them) indicating possession or close association.

There are many applications of a genitive case. It's a broad term and IIRC English has only 3.

English is a Germanic language, so it's a useful as a sister reference language.

Here's a better example of what is missing from our patios.
https://en.m.wikipedia.org/wiki/Genitive_case.
(Scroll down to the German portion)

Articles

The genitive singular definite article for masculine and neuter nouns is des, while the feminine and plural definite article is der. The indefinite articles are eines for masculine and neuter nouns, and einer for feminine and plural nouns (although the bare form cannot be used in the plural, it manifests in keiner, meiner, etc.).

Nouns

Singular masculine and neuter nouns of the strong declension in the genitive case are marked with -(e)s. Generally, one-syllable nouns favour the -es ending, and it is obligatory with nouns ending with a sibilant such as s or z. >Otherwise, a simple -s ending is usual.

Feminine and plural nouns remain uninflected:

• des Beitrags (of the contribution) – masculine
• der Blume (of the flower) – feminine
• des Landes (of the country) – neuter
• der Bäume (of the trees) – plural

Singular masculine nouns (and one neuter noun) of the weak declension are marked with an -(e)n (or rarely -(e)ns) ending in the genitive case:

• des Raben (of the raven) – masculine
• des Herzens (of the heart) – neuter

The list goes on and on for adjectives, etc.

The article also has a section for English.

Old English had a genitive case, which has left its mark in modern English in the form of the possessive ending 's (now sometimes referred to as the "Saxon genitive"), as well as possessive pronoun forms such as his, theirs, etc., and in certain words derived from adverbial genitives such as once and afterwards.