So, if you change company when your insurance comes up for renewal, make sure you make a point of telling your old insurer, preferably in writing and keep a copy of the letter and a note of who you speak to at your old insurer & when you did it.
That is your quote from your post.
so that is what you should have done. It works both ways. Say you forgot to "renew" your insurance on the exact date it was due and you had an accident and you were not insured. Your argument would be "I thiught it was renewed automatically".
If you know that you are going to cancel your insurance because of a new insurer then cancel your direct debit.