Going from Property Insurance to Renter Insurance

I just rented my first investment property which has a regular home owners hazard insurance. Now that is rented I want to change to Rental Insurance as it is cheaper ($800 vs. $500 per year) and also because it is designed for that purpose.

However, I am finding myself in a position where I need some advise. Here is the story.

When I applied for the loan I made sure I explained that I bought in the area considering that in the future we will use it as a second home, but that we will also consider the property as investment and we may rent it.

At one point I told the bank that the difference in financing between investment and second home was so low, that I rather do investment, that would leave me more options, but when the loan came out, it was a second home.

But now that I am about to change it to rental, a thought came to my mind...

If the property is second home it should have a home owners insurance, will changing it to rental be possible?

How do I change that? Shall I let the bank know that switched to investment property and we from there?

Thoughts?