What I object to is the idea that the fin is a separate structure from the bevel. To me there is only the bevel which extends all the way to the edge. To me that is the fin. But that is an entirely American definition.
I do thank you for providing the European definition. Maybe we should adopt that?

A burr is a consequence of a specific honing method that can be seen and felt with ease when sharpening various tools like plane blades, knives and chisels.
I do not believe that we develop a burr on razors when we hone using the standard alternating side honing stroke. I have never felt one. It may be possible to develop but I do not see a purpose in it.

A wire edge , on razors, is a consequence of excessive honing and is actually an edge/bevel that is to thin and is disintegrating.

OK, time for my second cup of coffee,