The word "sum-plectic" as a greek translation of "com-plexus" was needed also to differentiate the study of "complex geometry" (complex numbers etc) from the study of "complexes de droites" (the geometry of 'line complexes') where the Plücker coordinates are associated to a natural symplectic structure on the space of affines lines in any euclidean space.

The concept of "differential symplectic geometry" has been introduced I believe by J.-M. Souriau in is 1953 paper

@inproceedings{Sou53,
Author = {Jean-Marie Souriau},

Booktitle = {Coll. Int. CNRS},

Pages = {53},

Publisher = {CNRS, Strasbourg},

Title = {G{\'e}om{\'e}trie symplectique diff{\'e}rentielle. Applications.},

Year = {1953}}

He introduces there the concept of "Variétés isotropes saturées", called today "lagrangian manifolds", name given later by V.-I. Arnold.