Once your claim is reported to the insurance company, they may assign their own staff adjusters. Carriers don’t have incentive programs for adjusters to pay less than what the contract calls for. They do of course have a supervisor or auditor making sure they are paying market rates, replacing only damaged items; in short, not overpaying. Their main job is to get contractors paid for repairing your loss and get you reimbursed for your personal property, so they can move on to the next case they’ve been assigned.
Some companies use independent adjusters. It is cheaper not to have payroll between storms when things are quiet, so carriers use independents instead of staff, or just when they need to. Independent adjusters are vetted by the insurance company to make sure they understand what's actually covered by your insurance policy, as well as the building and trades business. They make sure contractors hired to make the repairs are charging a fair market rate for the work done and they are paid a fee by the insurance company for this service.
After a big weather event with lots of claims, many companies will hire independents to take up the backlog, so a combination approach is common when it's busy.
Public adjusters serve a different function. They are paid by the customer via a fee calculated off the total amount of the claim. This is good in the sense that their incentive is to collect as much money as possible from the insurance company so you're on the same side of the table. But you should be cautious when using a public adjuster, too: