Can either the entire or a portion of the plaintiff’s 33 1/3% attorney fees be covered by the defendant’s insurance damages paid if awarded?

Contingent attorney's fees in a personal injury case are a percentage of whatever is recovered.  They are not included in the amount recovered or the amount claimed by the plaintiff.  For example, if your settlement in a personal injury case is $15,000, first the attorney's 1/3 is deducted.  Attorney receives $5,000.  Then, medical bills where the doctor has a lien which means the doctor is paid out of the settlement  are deducted.  For example, if medical bills are $3,000, the doctor is paid.  Then, the balance would go to the client which would be $7,000.

If this had not been a contingency fee case and let's say the case resulted in judgment for the plaintiff,  plaintiff could then recover attorney's fees from the defendant.  If plaintiff lost and judgment was for the defendant, defendant could recover attorney's fees from the plaintiff.

In a non-contingency fee case, attorney's fees would most likely be an hourly rate.

The advantage to the plaintiff in a contingency fee case is that plaintiff doesn't pay a high hourly rate.  If plaintiff loses, plaintiff pays nothing other than perhaps expenses.  If the plaintiff wins, attorney is paid a percentage of the total recovered and plaintiff receives the balance after other costs are paid.