1.x communicates (to me at least) you are pretty happy with the current state of the package and don't see any considerable breaking changes in the future. When 2.x comes around, this is often after 1.x has been in use for a long time and people have raised some pain points that can only be addressed by breaking the API.
Because this comment, "The project is still in development, it might be stable enough for use in "real projects(tm)", but it might also still significantly change." That describes every project. Every project is always in development. Every project is stable until it isn't. And when it isn't, you bump the major number.
Nobody cares that Chrome's major version is 147.
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.
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.
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.