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What exactly is proved by the violation of Bell's inequality?

Although Bell published his inequality more than 50 years ago, and although it’s violation has been observed in experiments since more than 30 years, there still exists an amazing amount of confusion with regard to the exact meaning of this finding. Does it exclude hidden variables? Does it imply non-locality? Does it imply both, or neither of them? The confusion is mainly caused by the circumstance, that Bell’s inequality is part of a ‘proof by contradiction’, i. e. the inequality is derived from some basic assumptions, and it’s violation proves that at least one of the basic assumptions must be wrong. Actually Bell (respectively Clauser, Horn, Shimony, and Holt), Wigner, and Peres derived the inequalities, which are experimentally violated, from quite different basic assumptions. Consequently the violations of these inequalities imply quite