You've got it perfectly! Especially the bit about adaptations and other versions.
Consider the Nolanverse. Bruce Wayne actually has an arc there, with a beginning, middle and end. He creates the persona of Batman to be a symbol of hope for the people, hoping that his actions would shake people out of apathy and prompt a revolution against crime and corruption. And after a while, he succeeds, with Harvey Dent and Jim Gordon being able to clean up the city. In fact, his mission is complete and he does get to hang up the cape and cowl...except that he didn't get his happy ending because the woman he loved died, and had she lived, she wouldn't have been with him anyway. So in a sense, he's still stuck being Batman, because there's nothing else in his life, even though Batman isn't needed anymore. When Batman is needed once more, he jumps at the chance to get back into the game. He saves Gotham one last time, but this time round, he finally is able to move on with his life, with a new love, and leave the mantle of Batman to someone else.
Now that's not the kind of ending the mainstream DCU Batman would ever get...