These are all local issues and can carry widely from place to place. Some advice for the future, put the location in the title next time, i.e. "some city Rhode island permit question" so people familiar with that jurisdiction might see it and you can get meaningful advice.
Exactly.
Some places the EC doesn't even pull any kind of permit.
Some places the EC pulls permits for the electrical portion of the project.
Sometimes if EC is changed or even if there are multiple EC's doing different portions of the project - they each need their own permit for their portion of the project.
Here if I took over a project because owner fired the previous EC, I still need to file my own permit. Possibly can file for less portion of the project depending on where things are at when this change happened, but basically I must file a permit if I do any work that requires permits. The other guy's permit is his, and closing that out is between him and the AHJ. As long as someone else finishes the project and files their own permit it usually isn't a problem closing it out though.
Whoever is last one when EC's get changed is who ultimately needs to pass the final inspection.