There is no "one size fits all" solution for home defense.This is the bad advice:
While sniffed at by the experts, a .410-bore shotgun loaded with birdshot possesses more close-range stopping power than any popular handgun caliber, with far less danger of overpenetration. It is also much easier to operate and shoot accurately in high-stress situations than any handgun (which required well-practiced fine motor skills). The fact that this particular variant came with a laser-sight made it even more appropriate for these specific customers.Birdshot is for birds, small fragile flying critters with hollow bones. All you have to do ruin their flight dynamics and the fall will probably kill them. People are not birds.
While it can be deadly at close range, birdshot is a notoriously poor penetrator. Remember when our Vice President "peppered" that old man in the face and neck last year? Happens all the time. At short range birdshot can do more serious damage, but most of it is still superficial. If your attacker is wearing heavy winter clothing, that is additional layers the shot has to go through. Buck shot will generally penetrate the 12 inches you want in a self defense round, bird shot generally will not. Use the one that can hit important organs, not the one that hurts your target but will not incapacitate him.
My post after the hurricane insanity of last year still stands. About the only thing that has changed is that 7.62x39mm has started getting a lot more expensive recently. But I still say a shotgun is a good choice. If a .410 is all your shoulder can handle, then please load it with the biggest buckshot you can. Otherwise move up to 20 or 12 gauge. 00 or 000 buck might be overkill (slugs certainly will be), but #1 buck is a pretty good choice especially in a 12 gauge at short range.
UPDATE: In the comments, Steamboat Willy suggests .38s using standard pressure glaser safety slugs or wadcutters due to overpenetration concerns. Overpenetration is when the bullet either misses or goes through the bad guy and then goes into what is behind him (like your kid's room). This is obviously a bad thing.
The guys at the Box O' Truth have looking into this stuff. When they tested wallboard penetration they found that pistol and rifle rounds generally go through a lot of walls (more than 6 walls), but will have trouble if they actually hit structural members like pine boards and studs. In comparison glaser "safety" slugs and shotgun rounds generally only penetrate 3 walls (6 layers of sheetrock). So you get essentially the same overpenetration problem with the shotgun as with Willy's pistols and, trust me, the shotgun packs much more of a wallop.