This question is bordering on off topic, but to address the Apple relevant question….
Which JRE does Catalina require? Or does one just install JDK?'
Catalina doesn’t require a JRE. macOS has no need for a Java Runtime Environment out of the box. You would install a JRE/JDK based on your needs/requirements.
Now, since you mentioned LibreOffice, a quick look at their System Requirements for macOS tells you what you need:
Due to an issue, there are the following restrictions with Java: on macOS 10.10 and newer, JRE isn't found, JDK is required
You need the JDK.
OpenJDK is the Open Source implementation of Oracle’s Java Platform, Standard Edition. It’s the same thing as the fully licensed Java Platform. So, in short, it works with all Java apps including LibreOffice.