Since the first modern Olympics held/was held in 1896, this has marked the fourth time  that the Games have been completely affected by outside forces.

正如你所说，since可以是连词，也可以是介词，所以这里用过去分词作定语或用was held作谓语动词都是可以的。since从句移到句首，你就可以清楚地看出，that从句是the fourth time的定语从句。