For some reason I've just never liked Spider-Man. He comes off as a whiney, ignorant child that never seems to grow up or mature despite everything he goes through. I love a good coming of age story, but he just never seems to become an adult.
Jane Foster when she was the wielder of Mjolnir. Not for anything about her personally, but the fact that Thor was treated as a codename. It's the dude's actual name, it'd be like if Sam Wilson went around introducing himself as Steve Rogers when he took the Captain America mantle. It's happened a few other times like with Eric Masterson, but at least he had the excuse that for most of the time he used the name he and the actual Thor were sharing a body.
I think it's both, his name and his power. In Thor 1 when Odin sends Mjolnir to earth he whispers to it something like "May he who's been worthy possess the power of Thor".
I thought vision was able to lift the hammer because he wasn't a living being? At least I came to that conclusion because he never possessed the "power of Thor"
Vision is alive. His body is composed of living tissue woven together with the mind stone and vibranium. That whole speel by Stark arguing that vision could only lift the hammer in the same way an elevator would was him rationalizing why his creation was worthy but he wasn't. The whole point of the scene where vision lifted the hammer for the first time was to show that he could be trusted. Because at that point, almost everyone had their doubts.
So Thor has always been the title of the person in possession of the Hammer, he converts himself into Thor by grabbing the hammer, the movies then changed that because in the Marvel Ultimate universe it's different, but Jane Foster is from the original comics, where holding the hammer made you Thor, and she did exactly that in the 70s, just a couple of decades after Don Blake.
The thing is that, as you said, it's happened several times before. Beta Ray Bill, Red Norvell, Eric Masterson... it's been established for a long time that in the Marvel universe the title of Thor, God of Thunder, may be held by people who aren't Thor Odinson (and that he might occasionally lose it, though so far only temporarily, at least in the main continuity).
Marvel ran a miniseries called "Battleworld." Yadda yadda, Dr. Doom a single planet composed of all the different Marvel timelines. The police force controlling everything is the Thor Corps, which includes dozens of different iterations of Thor, including a Groot Thor.