Dana's dictionary of Nautical terms (1841-1851), as well as Lever's (1863-1870), speak of ropes in the definitions; e. g., "sheet is a rope used..., etc."
A search finds little use of the term 'line' by sailors of the 19th century.
I believe Strudelbag is corect. Chantey-singers and training schools are responsible for the current standardization upon 'line.'