1997 Grand Marquis - remote start

My folks have been complaining lately that it takes up to three manual attempts before the remote start kicks in. Three manual attempts translate into approximately nine overall tries since with each manual attempt the vehicle tries to automatically start three times before giving up. Not knowing where to start, I took it to O'Reilly's to do an "onboard" diagnostic of the battery, alternator and starter. All three passed, but the manager said that since the battery was still under warranty I may want to take it to the store where it was purchased and have them do a more thorough check. Long story short, the battery was bad and they replaced it.

I kept the car for 24 hours, and although it starts on the first manual attempt, it still takes 2-3 automatic attempts for the car to start (which is unusual). Could this be a sign that the starter is starting to fail? Are there any other warning signs or tests to determine when a starter is at the end of its life? It was replaced about 6-7 years ago, but given the amount of salt that is used in the Midwest on the roads in the winter time...

Thank you.

Update! The cable going to the starter is really corroded.