Answer
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According to author Lloyd Shaw, (Cowboy Dances and Round Dances, A Century of Waltzing), visitors from the French court to Elizabethan England observed the folk dances of the rural population in the north of the country and in Scotland and Wales.  These dances were performed by two lines of dancers facing each other, men in one, women in the other.  The speculation is that these dances were largely unchanged from Druid times, where they may have had religious as well as social significance.  Observing the two lines of opposing dancers, the French called them contre dances and took them back to the French court.   The English court then imported the dances back from France and kept the name the French had given them.  Except they corrupted the name.  Contre danses became Country dances.  In 1651, John Playfir published the first collection of country dances ever, The Country Dancing Master.  Contra dances survive to this day; the Virginia Reel is close to the originals.

Country music, while derivative in part from the music brought by Irish and Welsh immigrants to this country in the 19th century, is pretty much a 20th century musical form.  And in the case of the music, 'country' does indeed refer to the countryside and its folk.