united-kingdom (and likely everywhere else)
It's likely that your policy documents say you must tell the insurer if any information is incorrect.
Immediately tell an insurer about any mistakes in the policy. In such circumstances it doesn't matter how it happened so much as it is rectified as soon as possible. It might cost you more but it's imperative you make sure the insurer has no reason/excuse to 'void' (invalidate) the policy and to not pay out in the event of an accident.
Scenario:
Alice is insured for Social, Domestic and Pleasure and Commuting (SDP+C).
Bob is insured for SDP.
One day Bob has an accident on his commute. The insurer refuses to pay out - or reduces the pay-out - on the basis that Bob wasn't insured for commuting.
You don't want that to happen. Even if after complaining you eventually get a ruling wholly or partially in your favour, it will have cost you lots of time and stress.
(Also, if your policy is void then there are Road Traffic Act implications.)
The insurance policy is a contract between the consumer and the insurer.
The consumer provides information about the risks to be insured and the insurer agrees to insure against these risks in return for the price paid by the consumer.
Commuting usually occurs at peak times when the risk is considered to be worse than at non-peak times (and priced accordingly).
