docs: Improve READMEs for `apis` and `javy` crates

[saulecabrera]

Description of the change

In preparation to adding more APIs I decided to do a very light cleanup of some of the documentation in this crate. I opted to remove the documentation in the README to reduce the overhead of having to maintain the same documentation in multiple places.

Why am I making this change?

Checklist

• [x] I've updated the relevant CHANGELOG files if necessary. Changes to javy-cli and javy-core do not require updating CHANGELOG files.
• [x] I've updated the relevant crate versions if necessary. Versioning policy for library crates
• [x] I've updated documentation including crate documentation if necessary.

[saulecabrera]

I had the intention to do something along those lines, I decided that it's probably not worth the trouble because we probably don't want to include everything in the README in the crate-level documentation e.g. the ## Publishing to crates.io section and other README-specific sections that we might add in the future. Personally I find it more valuable to have great crate-level docs (showing up in docs.rs) than having some content in the README in crates.io.

[saulecabrera]

On a second thought, bringing just the example at least might be reasonable to do, I suspect that one won't need to be updated as frequently.

[saulecabrera]

Added the example, and I also realized just now that the example in lib.rs contained a snippet using the old bindings, so I also fixed that.

[saulecabrera]

Ok so got a bit inspired and improved the READMEs across the board they not only include a direct link to docs.rs but also the badges to crates.io and docs.rs. With these changes I don't think there's a chance of confusion around how to navigate to the crate-level documentation.