Unfortunately, you are not going to be able provide your insurance company with proof that the moving violation was reduced without notifying them of the inattentive driving infraction. The court signed an order reducting the moving violation to the inattentive driving infraction. You will need to obtain a copy of that order from the court to provide to your insurance company. The order will show the inattentive driving infraction. You may want to contact a local attorney and see if the inattentive driving infraction will affect your insurance rates.
Inattentive driving or inattention, as it is known in some jurisdictions, does not get reported to insurance companies because it does not go on the driver's abstract of driving record.
A collision might or might not raise your rates. The other thing that might raise your rates is a moving violation that is reportable. If the insurance company is raising your rates because you got a ticket - not because one was actually committed - I would urge you to shop for new insurance.
What you could provide the insurance company is a copy of your abstract of driving record or a copy of an insurance abstract (a more limited driving record).
Inattention to Driving is a Municipal Infraction, not a violation of a State Law. As a result it does not get reported on the three year report given to insurance companies by DOL. Your insurance company can order an updated report and they will see that the original violation no longer appears. You can do the same and show it to them. Your rates may also have been raised because of the accident and the infraction may have little or no affect. If your insurance company is unwilling to request a report from DOL and make the appropriate corrections you might want to consider a different company (there are LOTS to choose from).