Neville….
Unfortunately you are wrong in your assumptions.
I’m a family law attorney and dealing with non-Jews who have many juniors, IIs, IIIs, etc. I have to be up on this.
A junior is named for his father with exactly the same name. It is not enough that the first and family name be the same.
II and III are people named for the same person, not juniors.
So, if there is a John Doe and a grandchild is named for him, that grandchild is John Doe II. If II’s child is also named for original John Doe, the child is III, but instead if the child of II is named for II, the child will be a junior.
GW Bush does not carry his father’s second name…Herbert, thus he is not a junior.