Shotguns can work for defending yourself or home, but I prefer a rifle or carbine in anything in .357 Magnum or more powerful.
John, I have a .357 revolver -- and other handguns. I lack confidence in them in the scenario posited due to lack of practice, and that situation is unlikely to change. In addition, I regard handguns as marginallly competent backup weapons as compared to shotguns or rifles.
I have rifles .303, .308, 6.5, 7.65, 30.06, and .223. I can usually hit where I point --- at least when not being shot at. None is as handy in enclosed spaces as my Winchester 1300. They also tend to overpenetrate, and that is an issue because there are others living in this house and in another house within 250 feet. I do not plan to be firing towards those others, but . . . .
At the relevant distance (15 feet), the load I am using produces an adequate 2160 ft. lbs. of terminal energy that will produce a 3" hole in a 6"-thick telephone book. That would cause significant pain to someone wearing a Class III vest and a 3" through-and-through hole in anyone not armored (or armoured for our UK members). (The wife's 16 ga. is significantly less powerful.)
I believe reliance on "intimidation factor" in a firearm or knife is extremely dangerous and ill-advised. Don't expect to do anything other than telegraph that you're armed to an intruder.
We simply come from a different place. I fully intend to give notice. I know that I will shoot at a person if it comes to it, but hope not to do so. Given the dogs (inside) and the fact that I am an extremely light sleeper, I think my scenario is extremely likely. I will be calling out a warning from concealment. If that increases my risk, I am prepared to assume that additional risk. Everyone has to decide for themselves.
The police, by the way, may take up to ten minutes to arrive (station 6 mi. away). I need to take care of things for that period. So, Arya, we have an extremely reliable alarm (dogs) that summon an armed response with excellent knowledge of the "field of confluict" - us.