Hi. Everyone should evaluate their personal needs, budget, and skill level.
Essentially, the only reason anyone carries a handgun is ease of carry and concealment. For actual use, longarms (rifle and shotgun) are easier to aim, easier to control, and typically fire more powerful rounds.
Every state is different in law and application, but a "back against the wall" clause is typical. Almost anywhere in the United States, someone in their home with noplace to retreat, can use force against a potentially lethal threat.
"Intimidation factor" is foolish and I do not hear it mentioned by those whose teaching and skills I respect. I do not believe in announcing my location and status via audible weapons charging. I will not hold anyone at gunpoint by choice.
Are you asking because you're moving out of the UK, Andy? If so, a carbine with 16"-20" barrel, in any quickly operable platform, and firing anything from a heavy pistol cartridge (.357 Mag, .45 Colt, .44 Mag) up to a medium level rifle cartridge (.30-30, .308) would be a fine choice for most. My personal dedicated home protection firearm for under $500 would probably be an M1 Carbine or 12 gauge shotgun loaded with reduced recoil slugs. For novice shooters, I would not suggest the shotgun. There are lots of lever-actions that would nicely fill this role, and a used .30-30 will probably only run around $200.
John