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If a personal injury case progresses to the point where one of the disputing parties files a lawsuit, one of those 2 parties may get free legal help from his or her insurance company.

Would it be someone making a 1st party claim?

No, a policy holder can make a 1st party claim against his or her insurance company. That claimant does have the right to hire a personal injury attorney. That same attorney would charge the claimant/client a contingency fee.

Would it be someone that has chosen to make a 3rd party claim against the person that is responsible for an injury-causing accident?

No, if you decide to file a lawsuit against the person that you allege to be responsible for the accident that caused you to become injured, you should not expect any help from your insurance company. Your insurer would not furnish you with a personal injury lawyer in Los Banos. You would need to hire one. Still, that hired attorney would probably charge a contingency fee. That would mean that the attorney’s fee would be a percentage of your settlement money.

Suppose you hit another car on the road, and then you were sued by that other driver. Would your insurance company provide you with a free defense lawyer?

Yes, your insurance company would have a duty to provide you with legal counsel, so that you could enjoy a strong defense. Yet, insurance companies do not have to meet that obligation, if their policy holder has engaged in some form of extreme action, at the scene of the accident.
Why has the law provided for that exception? Because some people become overly aggressive, when trying to assert themselves at an accident scene. They take some sort of extreme action, in hopes of obtaining the possible compensation, or in hopes of getting revenge for the personal injury.
For instance, a driver that has been hit by another motorist might attack that same motorist. That would be an example of an extreme action, in that case one that was taken with thoughts of revenge.
Someone that has taken such an action has chosen to ignore the restraints of the law. The legal system does not look kindly on such a person. Therefore, if the same individual were to seek legal help from his or her insurance company, that would be denied.
The insurance company would appreciate the fact that it did not stand much of a chance for winning that court battle. Because the plaintiff had acted so foolishly, he or she would not have a strong case. In light of that fact, the insurance company would not want to spend money on a lawyer. Hence, it would seize on the available exception.