Historically there is no set version of a full english. What you describe is just your version. It will be, entirely, a social construct. This is why the full english varies so much, its different traditions in different areas and families being passed down, giving everyone a different vision of what it is.

Its similar to how everyone has their own christmas traditions, or how fish and chips in the north tend to be more traditionally served with gravy mushy peas and bread+butter. Whereas in the south, typically, they are sold with just ketchup or mayonnaise. But again, not exclusively. They only requirement is a fash and some chips. Everything else is just a variation of that but still counts.

This is why i belive that there only needs to be a few core ingredients for a full english to qualify as a full english. After that its all tradition and preference.