Hey @ mkstewartesq ! I never said that the Skywalker GRAFLEX resisted Kylo Ren because he wasn't a Skywalker descendant. I was implying that Rey, as a Skywalker, was able to call upon the lightsaber because she too had a claim to it. It came to her over Kylo because Kylo has succumbed to the Dark Side, and it's had enough of being wielded by a Dark Side user (Anakin).
Now, even if she isn't a Skywalker, I believe it went to Rey because she is a Light Side Force user and Kylo is a Dark Side Force user.
Other people have been able to use the lightsaber because it has a big red button on it for non-Force users to activate it. Remember too that Darth Vader ignighted Luke's ROTJ lightsaber, so you don't have to be a goodguy to ignite a goodguy's lightsaber, you just have to know where the power button is.
The saber in the snow scene on Starkiller Base was perhaps a bit more "Sword in the Stone" than the original scene in ESB as far as to who was going to win the tug-of-war with it, but I don't think it was too much more than showing that Rey was beginning to use her powers. Much the same way Luke was there on Hoth.
It also could simply be a tug-of-war scenario where Rey was the victor and the lightsaber didn't have any say in the matter. Rey was more powerful than Kylo in this situation. As she was in the torture chair scene earlier in the film.
Ultimately, Star Wars is a story devoted to the Skywalker family story. It makes perfect sense that if Rey is to be the hero in the coming movies, that she would be a Skywalker.