Summary: I have a wall between my neighbour and our property that has just been rebuilt and I have paid for a % of the costs to raise the height. I now want to ensure that any legal requirement are covered so that I am allowed to affix things to our side, and similarly there side. The questions is do I need to legally do anything do anything to "declare" the wall is a party wall?
Before I go further, let us get the jokes out of the way so they don't derail the thread:
Wall Party, Party Wall
More Info:

The land to the right of the fence use to be part of my property but was sold off in the 70's leaving a driveway outside our land.
To the right (red arrow), there use to be a low wall and a row of conifers, which gave us some protection from people walking over.
The tree's were not looked after, got covered in ivy and died, then fell over.
A pleasant chat with the neighbour led to a plan for a new wall, which I could pay the additional cost to raise to 0.9m to 2m. I would get permission to use our side for whatever we want to do.
At the base of the main wall, you can see a course of red, older brick. This is the boundary and entirely on my neighbours side.
However, the pillar (red arrow), has a brick width on our side.
I want to ensure that I can use that pillar for a gate but also use the main wall to support something like a carport. This will allow us to reduce the fencing on the left side and bring the driveway into the house plot, as it is currently isolated due to how I can't control the dog or kid escaping.
Before I go further, let us get the jokes out of the way so they don't derail the thread:
More Info:

The land to the right of the fence use to be part of my property but was sold off in the 70's leaving a driveway outside our land.
To the right (red arrow), there use to be a low wall and a row of conifers, which gave us some protection from people walking over.
The tree's were not looked after, got covered in ivy and died, then fell over.
A pleasant chat with the neighbour led to a plan for a new wall, which I could pay the additional cost to raise to 0.9m to 2m. I would get permission to use our side for whatever we want to do.
At the base of the main wall, you can see a course of red, older brick. This is the boundary and entirely on my neighbours side.
However, the pillar (red arrow), has a brick width on our side.
I want to ensure that I can use that pillar for a gate but also use the main wall to support something like a carport. This will allow us to reduce the fencing on the left side and bring the driveway into the house plot, as it is currently isolated due to how I can't control the dog or kid escaping.
- I believe the pillar is automatically a party wall as it is over the boundary (is this assumption correct?)
- The wall is entirely on their side, I can find examples of how I ask for permission to work on a party wall, but can't find how I declare the wall is a party wall.
- Or is that expliciet once I have permission to do work on it?
- I will then affix a bracket and some wood to use the permission