I'm joining the conversation late, so forgive me if I'm missing part of it or misunderstanding . . .
I would not consider "farmer" as a culture, but rather as a background. There are certainly, IRL, cultural elements to being a farmer different from being raised in a more urban environment, but it's not usually what we are referring to when we speak of culture. It's not one-to-one, but culture rides along pretty well with language.
I didn't say that farmer is a culture. Rather, a farmer (or rancher, or similar) is someone representative of agrarian culture to a relatively broad degree, and sufficiently representative (compared to a non-agrarian culture) that getting a +1 Str from it would be entirely believable simply because of what growing up in that culture means.
For example, if I were raised on a farm IRL, I might describe myself culturally as American, or perhaps some sort of hyphenated-American if I strongly identified with a subculture or immigrant culture.
You skipped some of my write-up, it seems. 'American' is a culture recognized in real life, even though if you ask for details, no one can really describe what it is. But there are some vaguely agreed-upon elements that you could derive from it, as a standard stereotype level. Same if you took any subgroup, such as the stereotype of rednecks, or African-Americans.
However, if I may re-quote myself:
First, do not conflate the Culture mechanic idea with the real term, 'culture'. A game mechanic is not the same as the real-life thing, even if we try to interpret it through that lens. It's the same as looking at Strength or Intelligence as reflective of their real-life counterparts. There are certainly similarities, and there's a broad parallel intent, but they are not the same thing. Making assertions based on the assumption that they are the same thing is arguing in bad faith.
And further:
As an abstract, that's fine. But how do you define individual instances? Dwarf culture vs Waterdeep culture vs nomadic culture?
For casual shorthand purposes, any of the above might work, but for the published rules you want to be as clean and abstract as possible. You don't want to reference Waterdeep, because what if you're running the campaign in Greyhawk? You don't want to reference dwarves, because what if your world doesn't have dwarves?
Even if D&D were set in modern day, "American Culture" would not be a useful mechanical descriptor. The term 'Culture' is used as a label for a mechanical element of the game which has a rough correspondence with what we might in casual conversation term 'culture', but is not the same thing. It's a label for a set of boosts (stats, skills, languages, whatever is decided upon) which are common to people who are members of a named group which can be agreed upon as being fairly narrow in scope, but broad enough to be easily understood by anyone referencing the rules. It also is a wrapper for social conventions common to that group.
It is not narrow enough to be a single person's occupation or choice of vocation. A culture must be broad enough that you can say that your character has a decently large number of things in common with other people of the same culture. A desk clerk who works in a farming town, but was raised in the city, and doesn't really gel with the other locals, isn't part of an agrarian Culture. He may be a member of an agrarian
society (and may equally describe himself as an "American", in your examples), but that is very much not the same thing.
Not everyone within the same community shares the same Culture. That is by intent.
Culture, as described by my writeup, is not the same as "Chinese immigrant culture" or "African-American culture" or whatever, because those appellations are vastly broader in scope (while also being extremely specific to the setting) than what can be contained within a mechanical write-up. Each Culture must be narrow enough that you can easily extract the intent of it, but broad enough that it can easily be applied to a number of places within your world setting.