The thing that seems to not be noticed or mentioned is that the fact that we may call it "just" Disney is because we are all basically lazy and adding two extra words is unnecessary. We know it is Walt Disney World. That fact has not left us, it is just that it is unnecessary to spell it all out. Even if we are talking to others we say Disney and if they want to know which one, they will ask and we will tell them. I also think that if anyone doesn't know that there are two Disney's in the US have been living under a rock.
The key thing is that there are still many people that still call Hollywood Studio's "Disney MGM Studio's" and MGM isn't even, now and never has been, a Disney property. So changing the name, plus every single thing that is listed as WDW would be useless and cost a lot of money to do. There is a reason why they call it "branding". Rebranding would be expensive, serve no purpose and be ignored anyway.