Good general summary, but these points are a little off....
Your definitions are wrong, check the wording of the '53 PCA. There are 3 criteria for off-weaps....
1. Any item
designed to be a weapon (eg sykes fairbairn)
2. Any item
adapted to be a weapon (eg pick-axe handle with nails in it)
3. Any item carried with
intent to use as a weapon.
Designed, adapted and intent. These are the
only 3 criteria for off-weaps (aside from items specifically listed as off-weaps per se). The first 2 are a done deal and easy. The last one is "intent" and that is the tricky one. It's "intent" that you fall foul of, when saying "i wuz carryin it fer self defence guvnor" - you have confessed intent. If you dont confess intent, then the police/CPS/Courts have to prove intent through circumstance, behaviour and context - it's a hard thing to prove what's on somebody's mind.
Off weaps most certainly do exist on private property, the same 3 criteria as explained above apply. However,
all the offences concerning the carry of off-weaps only apply to carriage in a public place. Again, check the wording of the '53 PCA for confirmation. There is no offence for having an off-weap in your home. You can have a baseball bat with 9 inch nails in it, it's definitely an offensive weapon, but as it is not in public, there is no offence. You can have a hammer by your bed and quite openly state that it is for self defence. It is an offensive weapon (by intent), but there is no crime because it is not in a public place. You can hang a sword on your wall - it's designed as a weapon and is a weapon by definition, but perfectly legal in your house because the law says it's only a crime to carry an offensive weapon in a public place.
It is not a crime to possess or carry an offensive weapon on private property.
Hope that helps.