People often use the words "agent" and "broker" interchangeably. Lawyers and judges often do, too. Although obtaining a policy by either an agent or broker may seem trivial, the distinction is one of major legal importance. And often times, the lines are blurred.
Under the traditional definitions, an insurance broker acts on behalf of the customer seeking insurance and shops for a policy among different companies with whom the broker has no official relationship. An insurance agent, on the other hand, works for a specific insurance carrier and has authority to act on the carrier's behalf. In the real world, however, brokers often have agency contracts with multiple carriers, and under the law, they may be considered "dual agent-brokers" who wear different hats for different tasks when helping the same customer.
Why does this all matter? In cases where the issues are focused on the actions of the agent/broker and not just the insurance company, the outcome will depend on this critical distinction. An insurance company may be held liable for its agent's conduct, but not for the broker's conduct. The insured may have a case against the broker, but not the insurer. If the case involves a dual agent-broker, the insurer might try to characterize the agent as a broker to avoid liability, yet the broker may be a less desirable litigation target for a variety of reasons.
In either case, the insured has a potential remedy against someone. The question is simply, who from? For example, an agent or broker who intentionally or negligently fails to procure insurance as requested by a customer will be liable to the customer for the uncovered loss. If the intentional or negligent procurement is a result of the agent, then the insurer will be liable as a result. However, if the policy was procured by a broker, the broker (not the insurer) may be liable for the resulting losses.
This is just one of many issues that can depend on this outcome. Other issues include representations about coverage under the policy; representations that the policy procured has sufficient limits; issues with representations on insurance applications that lead to claim denials; and more.
Answering the question -- agent or broker -- is a complex task that requires an analysis of all important facts, a comparison of those facts to legal precedent, and most often, sufficient expertise of the insurance field to properly argue which category the case falls under. What the agent says they are, or what the documentation they provide may say, is just one piece of this complex puzzle.