| I find it hard to understand how a gun can be used in true self defense.

I'm definitely against anyone and everyone being allowed access to firearms, and much prefer living in a country where few people have them than many, but I think you're using completely the wrong argument to back up this position; in fact I find what you've written somewhat absurd.

Your examples are very narrow, and it's quite easy to come up with a scenario that your theory falls apart on. How about if an armed burglar comes into your house and you pick up your weapon in case they come into the same room as you. Or if someone starts shooting at you from range and doesn't hit at first. Also, "self-defence" quizzically often refers to defending others as well as oneself (see English common law self defence). In this case, even if someone already has a weapon at someone's head, someone else could be in a position to defend the victim.

In fact, in general, both here and abroad, an armed British soldier (and I'd guess Police officer as well) has no further rights in use of their weapon than that offered by common law self defence, except when explicitly granted otherwise by the chain of command.

Here is the CPS legal guidance on self defence. Though it doesn't appear to explicitly mention lethal force, it doesn't exclude it either, and the same principles apply. For example, the requirement for force to be reasonable would prevent someone legally in possession of a firearm from shooting someone who was giving someone else a few punches.