I wouldn't worry about a car fax unless you're bored.
A friend had unlimited car fax so we ran my truck - I bought it new and had 2 insurance claims, one for a deer fixed at a licensed body shop fully paid by my insurance. The second an incident fixed at the Chevrolet dealer and paid in full by insurance.
Neither were on my car fax. Only thing showed was the last service at 30,000 miles at the same dealer that fixed the other damage.
The deer damage was in 2008, the other in July 2011, and we ran the carfax in Oct 2012