Home defense is often ranges at 10 feet or less.
At 10 feet, you could fill that shotgun shell with chicken feed and probably drop a man with it.
On a historical note, it might interest you to know that Bonnie Parker (Bonnie and Clyde the bank robbers) carried and used a cut down .20 gauge shotgun.
For home defense, get yourself an extra .20 gauge barrel and take it to the gunsmith. Have him cut that barrell back to 20 inches and round-over the crown after he makes his cut. If you want to, put the bead back on. You dont have to, but you might want it later for something.
Buy some Remington or Winchester #3 buckshot. That contains 20 heavy lead projectiles. At 10 or 20 feet, it is a seriously devastating loading. Another positive feature of .20 gauge buckshot is that it is easily slowed down or outright stopped by fixtures or walls in the house, so your chances of accidentally killing another occupant in the house or perhaps in a neighboring house are greatly reduced. If you load with a .20 gauge slug, it will punch through your interior walls and still be at a deadly velocity. In most cases, your exterior walls will stop the slug from reaching neighboring residences. That .20 gauge slug is a very serious manstopper at 20 feet. I dont think anyone is going to walk away from a belly hit with either the buck or slug at that distance. You dont need 12 gauge for home defense. You dont need a rifle or pistol for home defense. The 20 is absolutely the perfect HD weapon.
The gun that Lee Lapin mentions above is perfect for what you need.