There are so many similarities between the stories of Lazarus in Luke and John that you are right to recognise that there is evidence of copying. The usual scholarly position is that the story in Luke is original to Luke and that the author of John was inspired by this parable and by the Lukan story of Mary and Martha.
Even if the name 'Lazarus' were a later addition to Luke's parable, we would still have Mary and Martha, who appear only in Luke and John, in the first case as poor villagers (Luke 10:38-40, including that Martha alone was overworked serving the guests) who were friends of Jesus and in the second case as wealthy residents in the town of Bethany (John 11:1-2, cf 12:1 including that she anointed Jesus' feet with a pound of spikenard, very costly) and sisters of Lazarus. It would not be plausible to say that they also were added later to Luke's Gospel.
Keith L. Yoder, from University of Massachusetts, demonstrates that the relationship between these texts was was literary in origin and cannot be explained by shared oral traditions, here. He concludes with a finding that on literary grounds the relationship is best understood as running from Luke into John. Although it is theoretically possible, although unlikely, that Lazarus was added to Luke's parable, the name was certainly there before the author of John's Gospel wrote his account.