As nearly as I can determine, although the NYS&W was leased by the Erie, it was operated as part of the Erie Railroad, using Erie locomotives and cars, at least through the 1930's and -40's. The January 1934 Official Guide shows, in the Erie listing, under the heading ERIE RAILROAD,
CHICAGO & ERIE RAILROAD COMPANY
THE NEW JERSEY & NEW YORK RAILROAD COMPANY
NEW YORK, SUSQUEHANNA & WESTERN RAILROAD COMPANY
BATH & HAMMONDSPORT RAILROAD COMPANY
In the schedule pages, the timetables of the NYS&W, New Jersey & New York, and Northern Railroad of New Jersey are all separately identified, each with the note "For List of Officials, see Erie R. R. pages 432-433." I know that trains on the the Northern RR and the NJ&NY used Erie equipment and I always understood that the same was true of the NYS&W until it became independent in 1940. I believe it was the trustee, Walter Kidde, who adopted the maroon and gray livery (later solid maroon on the Stillwells), as well as the whole Susquehanna Transfer concept. So it appears that until the separation the NYS&W was operated as an integral part of the Erie.