The movie is very much based on the idea that Charles though old has touched so many hearts and logan as well no matter what wrong he has placed before him he has always put mutant life ahead of so many others. Logan does resent him but even though that is true he still is the only person that understood logan so well.

That would be cool that he doesn't call them mutants. Mistique would like that as well it be fun for her to think about hersaelf being a hero more then something the world hates.