Firstly, country music isn't exactly 'North American white folk music' - it's more 'North American Southern rural white pop music'; 'country music' is explicitly the music that came after record companies realised that there was money to be made in marketing versions of (white) Appalachian folk music and other related styles, in the wake of the runaway success of Ralph Peer's discoveries of Jimmie Rodgers and the Carter Family at a recording session in Bristol, Tennessee in the late 1920s. Over time this music fundamentally changed from scattershot recordings of lightly-altered traditional folk music to a commercial music industry most prominently located in Nashville. A particular songwriting style emerged within country music, one which was based in Tin Pan Alley commercial songwriting (perhaps filtered through blackface minstrel shows) as much it was based on traditional folk - at some point in the 1930s/1940s, country music stopped being about traditional folk tunes and started being primarily about new songs - pop songs, effectively - just in a country style (and names like Roy Acuff and Hank Williams along others are usually given some credit for codifying aspects of that songwriting style).

In terms of why it's called 'country music', well, for a very long time from the late 1920s to the late 1940s, this particular genre of music (as distinct from the Appalachian folk that was obviously a big influence on it) was not commonly called 'country music'. Instead, within the music industry, such music was often called 'hillbilly music'. For example, a 1944 Billboard article about the success of country music - pointing out that Jimmie Rodgers sold more records than the opera singer Caruso - refers to 'hillbilly and cowboy music' where we would ordinarily expect to see the phrasing 'country and western'. Billboard magazine until 1949 has a 'Most Played Juke-Box Folk Records' chart, which chronicled the 'hillbilly records most played in juke boxes', and featured names like Eddy Arnold and Tex Williams.

At some point in 1949, however, the genre descriptor in the Billboard charts referring to this kind of music changed from 'Folk' to 'Country & Western (Folk)'. This is because, basically, the music industry wished to rebrand the genre so that instead of being called 'hillbilly' - not exactly a flattering description of rural white people - 'country music' was the preferred description, in the probably accurate belief that more people would buy 'country' music rather than 'hillbilly' music. This change of name, more or less, coincided with the mainstream commercial record industry putting down roots in Nashville, Tennessee, opening offices and studios in the town that were specialised in serving that particular genre of music (whatever it was called). In other words the label 'country music' is a label that was trying to symbolise a certain level of respectability - it's not just hillbillies playing old fashioned hick tunes, but it's a genre style that could be appealing even to urban audiences, that the record companies had put a long term investment into (the Nashville studios) and expected to make money from.

So it's not quite either about the increasing differences between the white and black folk traditions (that were always linked but somewhat separate) - the term 'country music' happened after that - nor was it a reaction to rock'n'roll - 'country music' happened before that. Instead, the term is more a reaction to commercial pressures - how, wondered the record industry, do we attempt to appeal to people who might turn their noses up at 'hillbilly music' but otherwise would be receptive to the sound?