“The word used in Greek is not cousin.”
Which “Greek” is that? Do you know the difference?
The word used in Luke 1:36 is a koine Greek word sugennes. It means that Mary and Elizabeth were somehow related; that relationship could have been by blood or by ethnicity or whatever. It indicates kinship of unspecified type. The idea that Mary was a first cousin to Elizabeth is a 3rd century, extra-biblical fabircation by Hyppolitus.