Gender is, as mentioned in your first quote, a system of agreement based on noun classes. Gender systems always have a semantic component (or in other words, there is a "semantic core" to at least one of the noun classes/genders): there are languages where gender corresponds to phonological or morphological criteria for most words--or for most non-human words specifically--but there are no languages with a gender system that is always arbitrary from a semantic point of view and based purely on phonology (Kibort & Corbett).
The trigger of gender agreement is expected to be the noun itself, not the phonological form of the first word after the article in the noun phrase.
The English use of a vs. an does not qualify as a gender system. It is not based on noun classes (as shown by my examples below, the same nouns, contribution and idea can be used with either a or an depending on what word comes after the article), and there is no "semantic core" to the phrases that take a or the phrases that take an.
The a/an distinction is phonologically (not morphologically) conditioned: as you said, an is used before vowel-initial words (not nouns--the only thing that's relevant is the pronunciation of the directly following word, which is not a noun in phrases like an important contribution or a good idea). From a historical perspective, this phonological conditioning comes from earlier differences in the rate of dropping the sound /n/ before vowels vs. before consonants: the /n/ was originally present in the article in all contexts.
I haven't heard of a language where a gender system originated from a phonologically conditioned sound change like this. However, in languages that have preexisting gender systems, phonologically conditioned sound changes can influence the gender category of nouns with a certain phonological form. For example, many Romance languages have articles that take different forms conditioned both on the gender of the associated noun and the phonological form of the following word. I know of some evidence that a form that was originally conditioned phonologically may be reinterpreted as indicating gender information instead.
Spanish nouns like agua
A specific example: in Spanish, the noun agua, which has the typically feminine ending -a and which consistently triggers feminine agreement on postposed adjectives, takes the definite article el, which is identical in form to the article used for masculine singular nouns, and different from the article used for consonant-initial feminine singular nouns (la). This use of el seems to have originated as a morpho-phonologically conditioned variant form of the article (el is used before nouns starting with a stressed /a/, whether masculine or feminine), and for the standardized form of Spanish, the rule is often described that way. (Wiktionary specifies that unlike the English a vs. an rule, the standard la vs. el rule is conditioned by both phonology and morphology, not purely by phonology, as it seems that la is used before feminine adjectives starting with stressed /a/.)
But ukemi wrote an interesting answer to this question on Spanish SE that indicates that for some speakers, gender may be involved, as apparently there are examples with such nouns of el being used even when a preposed adjective intervenes between article and the noun, and of preposed adjectives being used in the masculine form.
It seems that, since the morphophonological condition for using el before a feminine noun is so restricted (when I first wrote this post, I misremembered it as applying to all feminine words starting with a stressed vowel, but it seems that it only applies to feminine nouns starting with the stressed vowel /a/ specifically, and there are exceptions even for the small number of words in that category), the phonological basis for the use of el with agua is not very salient to Spanish speakers, so the form el is prone to being reinterpreted as an example of masculine agreement for words preceding this noun.
(One thing I'm not sure about is to what extent the confusion/variability in agreement for adjectives and other preposed words before nouns like this is a matter of spelling vs. pronunciation. One of the examples in ukemi's post is que el abundante agua caída; my understanding is that because Spanish speakers blend vowels together between words, this would when spoken sound very similar or maybe even identical to the standard form que la abundante agua caída. But the example por aquelm asa y no por éstaf is clearly distinct in pronunciation from the form with feminine agreement (which would be por aquellaf asa y no por éstaf) as l and ll represent different consonant sounds in Spanish.)