I was responding to your statement that "According to this Times article, to reach the Giants/Jets stadium from Manhattan, you have to change from the DL&W to the ERIE at Secaucus."
From what I can see, the NY Times article doesn't mention predecessor railroads at all, it just has a link to the NJ Transit web pages.
Former DL&W and former Erie tracks are involved, but not as you stated. Coming from New York, to get to Meadowlands stadium, at Secaucus Junction one transfers from a train arriving on former PRR tracks to a train on former DL&W tracks. Back in the day, I don't recall the Erie, DL&W, or PRR having any trackage rights on each other's rails from the site of the current Secaucus Junction to Meadowlands stadium. (Around 1956, a few years before the Erie and the Lackawanna merged, the Erie started using the DL&W Hoboken terminal station on the Hudson River for its passenger trains instead of its own Pavonia station, but the connection between the Erie and the Lackawanna was farther towards the Hudson River (compass southeast) than the modern site of Secaucus Junction.
J S