There are JLS (and JVMS) defining the "core" java. There are also a lot of JSRs extending its functionality. But is there a specifications for Java's standard library like java.lang?

The closest I found is Java™ Platform, Standard Edition 8 API Specification, but it feels more like documentation rather than specification. Is this doc the official spec for java standard library APIs? Do vendors other than oracle (like AdoptOpenJdk, etc) read this doc-like spec and implement their version of JRE? Or there is another more "formal/official" spec?