Your insurance company will notify your lender of your landlord insurance policy but this will not create an issue for you and the lender will not require you to refinance your mortgage. As long as you were truthful and accurate when you submitted your original mortgage application you will be fine.
Based on your question, it is pretty clear that the property was properly classified as a second home when you bought it which means you correctly obtained an owner-occupied mortgage. Most important, the home has not been rented out since you bought it. You should feel comfortable renting out the home now without the risk that your lender will require you to refinance your mortgage.
Please note that your situation is somewhat common as the occupancy status of homes changes for many borrowers after their mortgage closes. For example, after living in the property for several years some borrowers decide to move out of and rent a home they purchased with an owner-occupied mortgage. On the other hand, some borrowers decide to move into a home they had financed with a non-owner occupied mortgage after initially renting the property. In neither case does the lender require borrowers to refinance their mortgage due to the change in property occupancy status.