It appears I was partially wrong - when parked at home, the liability is covered under the home insurance, when parked at a campground, the liability is probably not covered under the home insurance, and when being pulled down the road, it's covered under the auto liability.
Bottom line is still the same - determine what you want protection for, see if it's included in your existing policies, and if not, start shopping.
https://www.insuramatch.com/blog/2019/04/my-camper-covered-homeowners-insurance#:~:text=When%20an%20RV%20is%20sitting,typically%20cover%20the%20associated%20expenses.