A simple test: ask whose side the producer is on. An agent sits on the insurer's side of the table. A broker sits on the client's side.
The definitions:
Agent (Ins. Code §31): transacts insurance on behalf of an insurer.
Broker (Ins. Code §33): transacts insurance with, but not on behalf of, an insurer. In practice the broker acts for the client.
One license can act either way, so §1623 sets a presumption. A licensee who maintains the required broker's bond and has a signed written disclosure telling the client she acts for the client is presumed to be a broker. In the admitted market that presumption is rebutted if she is appointed as the insurer's agent for that same class of insurance. The disclosure alone does not decide her status.
Duties follow the side. An agent owes the insurer loyalty and a duty to report what matters. Under Civil Code §2332, as against a principal, principal and agent are each deemed to have notice of what either knows and ought in good faith to tell the other. So if an applicant tells the insurer's agent about a home daycare and the agent leaves it off the application, the insurer is generally deemed to have notice of it.