By releasing a library with version 1.0, I communicate: "I consider this project to be in a state where it is reasonable to depend on it".
By releasing a library with version 0.x, I communicate: "I consider this project to be under initial development and would advice people not to depend on in unless you want to participate in its initial development".
I don't understand why people find this difficult or controversial.
There is additional subtlety here.
For example, sometimes projects that have a 0.y version get depended on a lot, and so moving to 1.0.0 can be super painful. This is the case with the libc crate in Rust, which the 0.1.0 -> 0.2.0 transition was super painful for the ecosystem. Even though it should be a 1.0.0 crate, it is not, because the pain of causing an ecosystem split isn't considered to be worth the version number change.
Oh hey I recently saw a comment which discussed this exact issue: https://news.ycombinator.com/item?id=47752915
99% of the time this situation is okay, because Cargo allows you to have both 0.1 and 0.2 in the same project as dependencies. It's just packages that call out to external dependencies, like libc, where it enforces the single version rule.
You can have both 0.1 and 0.2 in the same project, but you really don't want to.
Most of the time, it works so well people don't even notice.
The only time you run into a problem is if you try and use values with a type from 0.1 with a function that takes a 0.2 as an argument, or whatever. Then you get a type error.