The restoration company suggested a public adjuster. Is that the same role as the contractor or insurer's adjuster?

No. The contractor documents and performs repair work. An insurer's staff or independent adjuster handles the claim for the insurance company; a public adjuster is separately hired to represent you in preparing and settling it. Hiring one is optional. Before signing, verify the required license and examine services, fee calculation, cancellation terms and how you will receive updates. Illinois guidance says public-adjuster fees are your responsibility, not a covered policy expense. Consider the likely benefit after fees; neither hiring a public adjuster nor using a particular contractor guarantees more coverage.