You're right, Jacques, it is very clear.
Which is probably why the NBTHK, Mr. Tanobe, and Kanzan Sato all agree with each other. And don't agree with you.
First, Sato writes that "nagamaki" refers to the way these swords were mounted. Followed by "Therefore, nagamaki should properly be called naginata." Which blows the copies claim theory out of the water. Why? Because despite the marked differences in shape, they are still "naginata."
Further, anyone reading Sato's article should realize that Sato is talking about blades made over multiple time periods encompassing a number of different shapes. To that point, when being reconfigured to other uses, katana, wakizashi, modifications would be adapted and customized to the individual sword. It was not a one cut fits all. Which accounts for the differences we now see in the boshi between the different types of naginata-naoshi.
On the subject of "impossible to know." Again, Jacques, you are mistaken. When naginata-naoshi underwent a shape change there was a narrowing. That narrowing changed the appearance of the sword. When properly restored by today's polishers, excellent polishers will recognize this distortion and reconfigure the shinogi to give the sword more of the original appearance, as the maker intended.
When it comes to nihonto, it's just when you begin to think you know something that you find out how little you know.