I think your memory is mostly right. Leto II essentially wanted to instill into humanity this lesson, do not fall for the words of the prophet, permanently: his reign would forever be remembered as the hell that this can lead to. And yes, he was satisfied with being killed, as this was the goal of his plan as well: torture humanity for a thousand years, teaching them a lesson they will never forget (especially in the universe of Dune, where genetic memory is a thing), and then be defeated by them, teaching also that no evil is too big to overcome. Note that this is about any prophet, false
or true. Both Paul and Leto II were true prophets, but this made them no less dangerous.
But Paul is exactly the failed version of Leto II. Paul saw the option of doing what his son did, but didn't have the courage to do it. He instead chose to believe that he can become the benevolent dictator that would uplift humanity through his strong moral core, and instead brought only needless suffering to the galaxy and to all those around him.
The good future that Leto II fought for is one where every human is truly unique and free, spreading in a thousand distinct directions, impossible to predict for anyone (which also implies that they are impossible to lead as an overall group, as they will never again accept to follow someone else).