Does commercial film insurance protect the production company or the client?

It protects both. It protects the production companies for claims that may arise from their legal liability and damage to equipment, sets and props.

If you purchase media production or commercial producers’ indemnity, the policy is also providing essential protection for your client, as it is covering their investment / budget.

The protection for the client is by virtue of the fact that the policy includes re-shoot costs. If things beyond your control go wrong, the policy will cover the re-shoot costs without the client having to pay twice! Please remember the cover for re-shoot depends on which product is taken, see Media production vs Commercial producers indemnity here.