So I guess Kitty was not Shadowcat because she was never called Shadowcat.

You are going too far with your argument. When it comes to characters, whether they don't use the codename, Selena will always be Catwoman even if they don't mention her name. But with team of superheroes, its NOT!
thats exactly the point I was going to adress after reading that Catwoman comment.
We arent talking about what each character is and what not, we are talking about who were the X-men members.
And
When the real x-men are formed.
On FC, Charles wasnt a proffesor yet, so if we take that into account, technically he wasnt Proffesor X yet, and the x-men who stayed with him at the end should be considered the very first x-men.
Angel and Darwin never were x-men, thats for sure. That's what some haters use to complain about on the net, but they never were x-men, so its not an argument anymore.
Magneto, Mystique and Moira worked with Charles, Alex, Sean and Hank, so they were a team working together since the manssion until the battle on the beach.
The key question is: When the
x-men were formed?
on the manssion? on the beach, or later on the school, when Alex, Sean and Hank decided to stay there?
That's the key