Mine is the anti-derivative, as i can use integration to find moments, volumes, lengths, probabilities etc. None of which are areas.

When looking for a function to describe area, you define an area between two bounds and find a limit that essentially leads to a differential equation that you solve with integration. Not all integration describes an area, but it is necessary to use integration to describe the area function.

I disagree. Every derivation of integration to find an area or volume starts with a squeeze function, that leads to a differential as a limit. Integration is then used to find the function that describes the area or volume. I always found the length of a curve frustrating as it lacks a squeeze.

Sigma and Pi are universally recognised as sum and product so don't require further definition. There is also a problem in maths education that the look and feel of A level maths is completely different to even 1st year undergraduate level, making the transition extremely hard for students.

I think it is the opposite. The whole point of notation is to accurately communicate an idea, and if that idea is complex then so will be the notation. As an analogy, you wouldn't want your best literature students to only read abridge versions of novels, purely because they are easier to access.

I would argue that notation, specifically, has to be pedantic. Ambiguous notation is no good to anyone.
After striking your row and column, do you mean cross product? Or determinant of the 2x2? As they are very different things
The triple scalar approach is my favourite.

I've got no problem with the phrase nth term, I just wish they'd occasionally ask for the pth, jth or fth term, at least it might make the student consider what the words really mean.