Country of course is a very broad term.

Early 'country' artists (Carter Family, Uncle Dave Macon, Fiddlin' John Carson, Eck Robertson, etc. the list goes on and on) often were lumped together as 'Hillbilly' - this music often had a strong traditional foundation (stronger in some artists than others); other sources and influences included mintstrelsy, vaudeville, music hall ('the stage' - all three) as well as nineteenth-century sheet music and other popular sources.

'Cowboy' music is a related yet whole other animal - the quintessential cowboy singer was Gene Autry, who got his start as an emulator of Jimmy Rodgers and went on to make numerous movies and records, riding on horseback and singing about wide open spaces and stuff like that. While 'Cowboy' singers did traditional material as well (ballads like 'Zebra Dun' and 'Sam Bass', etc.), much more of this stuff was made up out of whole cloth. Autry and other cowboy singers usually dressed up in flashy, faux Western garb while riding up the canyon as the coyotes yipped. No clear delineation between Hillbilly and Cowboy, of course - many artists did a little of both. To simplify, Hillbilly preceded Cowboy and had a stronger traditional foundation (not that real cowboys didn't sing songs, but Cowboy as a commercial genre came later).

The honky-tonk country music of the 1940s, 1950s and beyond (Hank Williams, Sr. etc) is closer to what most people would unquestionably identify as country music. It has a clear blues influences and generally fewer specifically 'Western' themes.

Nashville was the first home of country music, but the Bakersfield scene in California passed it by in the 1960s and 1970s (my opinion, of course). The 'Okie' migration of the 1930s and 1940s provided an essential import of music and population from Missouri, Arkansas, Texas and Oklahoma (Maddox Brothers + Rose, etc.) The West Coast scene took off in the second half of the twentieth century (Merle Haggard Buck Owens, etc.).