My HD weapon, . . . the one I am carrying today. On this day, it is my Besa Thunder .380, . . . but normally is a 1911 in .45ACP.
Why the handgun? HD encompasses so many scenarios, . . . and many do not lend themselves well to a shotty or even a carbine (try answering your front door with an 870 in hand, . . . and the visitor not thinking you are somewhat spacey).
HD does not always offer one the time to pick and choose the time and/or place in the house when the "attack" will come, . . . necessitating a weapon that can be accessed quickly and deliberately, . . . and put into action on a short moments notice.
HD also often requires the homeowner to be very careful of the weapon employed, . . . as rounds streaming through exterior walls or windows will land somewhere, . . . and in your neighbor's house is not the desired place for them to quit their flight, . . . much less in an innocent bystander outside your house. Most handgun rounds will also be more likely to stay inside the building than rifle rounds or slugs.
Finally, . . . there is never very much in the average house that will fully provide "cover" for any bg in your house, other than a full refrigerator, freezer, or some wood stoves. There really is no need for the extra penetration given for the rifle or hopped up shotty.
IMHO
May God bless,
Dwight