Wow! what aloaded question seams simple enough until you answer it.
{"What is meant by "European Style" ballroom dancing? Also what are dancers called when they compete in each dance of each classification?"}
what context? Most dances have been "defined/sterilized" for competition by european countries English Watlz of the French original, Austrian Vienesse Waltz, the British version of Tango or Cha Cha.
What is style? well style is how you move over your feet. with "International" style Latin the body is hel d very erect with the emphasis on technicality in "American" style latin the same dance is characterized by more hip and leg action giving a greater erathier motion. In the clubs of latino culture the style is one of laise affaire very laid back with almost no sense of posture. Until showtime makes them perform.
Now what do you call a dancer? when amateur is with a teacher this is called pro/am when dancing with another who loves to dance they are called an amateur couple when they receive any form of compensation example: money, housing, or food they are called professionals in America. Europe has a more liberal definition tha I disagree with. When the dance is ballet they can be refered to as a balletomaine.
So we go back to what context. sorry to confuse but I hope to help clarify the question.j.0

j oo–