I don't think you've been misinformed, I personally think the marketing of the 20% time idea is better than its actual implementation.
Pitching the 20% time ideas to management is kinda killing the concept. Managers aren't particularly inclined to incorporate hours into your schedule, especially if that time could be used boosting their metrics.
The implementation of 20% time is incredibly variable across teams, and even across individuals. I've never had to pitch a 20% project to my manager, or even notify them that I'm working on it. For many past ideas I ran it by them just because they might be aware of people that can help me or related projects I might want to look at, but for my current 20% project (an eventually open-sourced library), I didn't even mention it to him until I went up for promotion, when a peer wrote "Oh, by the way, his 20% project will have a big impact both on our team and on the world at large", at which point my manager was like "Maybe we should make this an official OKR and part of your regular job duties." That actually happens a lot - I've had a bunch of 20% projects that I mention to management and they're like "Oh, you should feel free to take some 80% time to finish that."
I also find that the definition of 20% time varies a lot between individuals. For me, it's "All that time when I have nothing better to do, when I get to work on whatever I want" - in some weeks, that can be 80%, in other weeks it's 0%. I have a friend who spends one morning a week working at Google Research, and that's his 20% project. I have another friend that taught Lego Mindstorms to 5th graders one day a week, and that was her 20% project. I have another coworker that started working on a new project (sponsored by a different VP), and that was his 20% time. I have a few coworkers that basically do whatever they want - actually, my job description is pretty much like this now - and so it's all 20% time.
Actually, for a lot of my coworkers, the reality is probably much like it is in many other large organizations with decent management: "As long as you get your work done and are aware of the organization's priorities, you can do whatever you want." I know someone who moved to Uganda without telling his manager and regularly works from Paris, Thailand, Budapest, Vienna, Prague, etc. - his manager doesn't care, because he's responsive to e-mail and gets his work done quickly.
I don't think you've been misinformed, I personally think the marketing of the 20% time idea is better than its actual implementation.
Pitching the 20% time ideas to management is kinda killing the concept. Managers aren't particularly inclined to incorporate hours into your schedule, especially if that time could be used boosting their metrics.