The duty to defend is broader than the duty to indemnify. But it is not without limits.
The duty to defend is created by the policy language. If the claim against the insured cannot be within the policy’s coverage, then there is no duty to defend.
I once represented a liability insurer whose insured had been sued. In the complaint, the claimant/plaintiff alleged that he had been “viciously assaulted” by the insured. In fact, the insured had struck the claimant in the back of the head with a tire iron.
But, the claimant wanted to trigger coverage because the insured had no money. By the 4th amended complaint, the allegation was that insured “negligently altercated” with the claimant.
Many jurisdictions would still hold that there is no duty even to defend in this case. I recall reading one opinion several years back where the court specifically held, as a matter of law, that a closed first punch was an intentional act. I’ve seen multiple opinions finding the same about gunshots.
The duty is triggered by the factual allegations of the complaint. Creative characterizations won’t overcome hard facts.