Duty to Defend means the liability carrier, your insurance company, must provide claims investigation resources and legal counsel to you in the event of a potential loss covered under any liability policy.
Implied in this duty is that the duty to defend is more broad than the duty to fairly pay legitimate claims. Since you must defend non-legitimate claims, it's the potential coverage which initiates defense, not the final indemnification amount.
What's not so clear is where this line is crossed. When does the insurance company not have a duty to defend?
Each state is currently struggling with this issue. The insurance companies are less inclined to pay legal fees for feud lawsuits or clearly excluded circumstances. Contractual liability, environmental and cyber liability seem to be the areas of test cases.
In the settlement negotiation process; however, the insurance company representatives are in a unique position to sway the proximate cause of the claim. For example, in a situation where a liability policy has an absolute environmental exclusion, but products are covered. If a claim occurs for the product creating an environmental issue, either liability could respond. In the case where the insurance company attorneys argue that the environmental liability is the correct one, and the company should not be responsible for legal fees, a conflict of interest arises. Should two sets of attorneys represent the defendant?
Obviously, this type of settlement where the company negotiates an uncovered peril as the cause is bad faith, and the public needs protection as the insurance company is protected against fraudulent claims like arson.
It is not, however, without precedent. Look at the uninsured motorist coverage. You pay for this coverage in your personal automobile or in the company fleet policy. An accident occurs and the other driver flees the scene. When you turn in the uninsured motorist claim, your insurance company hires a lawyer to defend the uninsured motorist, the hit and run driver, against your claim. On the surface, this protocol may not seem logical; but the uninsured motorist may be caught. In order for the insurance company to collect through subrogation, they must demonstrate mitigation of the loss, that they acted in good faith on behalf of the other driver or car owner.
Under professional liability policies, the insured pays a first dollar retention which includes the cost of investigation and indemnification. After that amount is exhausted, the insurance company pays for all legal costs until they reach a settlement agreement. At that point, the insured has the option of accepting the agreed terms or taking over the claims procedure and paying any legal fees over the settlement amount. Or, any additional indemnification awarded.
It's worth researching your state law regarding legal expenses and the duty to defend. This issue will be settled on a state by state basis.