I wonder how these specific insurances compare to coverages under something like an umbrella policy. I don't have any special firearm insurance other than an add on policy for the value of the guns under my homeowners policy. I had considered getting an umbrella policy but never got around to it.
If I get around to calling back my insurance agent I will ask and see what they say.
I don't know what you think an "umbrella policy" refers to. I've learned not to assume that you have the correct 411.
Homeowners policies, and most renters policies, EXCLUDE intentional acts that cause injury. In other words, your policy covers accidental injury to visitors, like slipping on your wet or icy steps, being hit in the head by a falling tree limb, and so on.
Accidents.
Shooting someone, regardless of legal justification, is an intentional act. it's one of the main reasons firearm use insurance exists. It covers both intentional and accidental consequences of firing a gun.
For instance, if someone is at your door breaking in, and you fire as the door slams open, one of those rounds will likely miss. If you damage the neighbor's property (or unfortunately hit the neighbor), the insurance covers that, too. A homeowners policy will not pay out ... intentional.