Sorry but I disagree. They are not the same. But you are entitled to your own opinion. Chicken or the egg?.. Apple and oranges
Both are not correct. It's called a doubled die because the design is doubled. That is the correct terminology, period! A layman can't add terminology to a field unless those in the field except it, which they don't.
The egg did and the chicken is still angry about it. Birds evolved from dinosaurs which were laying eggs long before any developed feathers.
In this case the DIE was an error, but the coins struck from it are a variety. Impressive but not legendary, because no one knows about it unless they've visited the Superintendents office. The 43 copper cent is legendary because pretty much every collector knows about it and so do a very large number of non-collectors Double was correct at one time, but the language has improved as the understanding of how it was created was better understood and the correct term now is doubled die. When the 1955 DDO cents were first discovered the standard explanation was that they had been double struck. As it was learned/realized that it was struck just once and that it was the die that had the double image it evolved into doubled die. So it evolved from Double die (struck twice by the die) to doubled die (struck by a die that had a doubled image). And the egg came first. Eggs were being laid LONG before the first chicken ever came along.