I do the vast, vast majority of my development and testing on the OpenJDK. Lucene and Elasticsearch are heavily tested on it and the OracleJDK. Both are safe to run Elasticsearch so long as you follow the versions in the support matrix.
I suggest using the JDK rather than the JRE for the same reasons @Joshua_Rich does: it comes with a bunch of debugging tools like jstack that are useful. The last time I checked those didn't come with the JRE. I suspect this is why the JDK is recommended though I wasn't involved in writing that page.
It looks like openjdk doesn't ship binaries for windows. I suspect they are tied to the linux build tools. I can't advise on distributing the oracle jdk either. Sorry!