Officially supported means it's actually owned by some team. They'll dedicate resources to it, meaning it will be accounted for on any project planning or resource management the management needs to do.

Officially supported things tend to be bigger.

If the project is officially open-sourced, that will be taken into consideration when project priorities, re-orgs, and direction occur at higher levels. "How does this effect our commitment to the community" is a question to address. If it isn't official, then is best-effort by the people who pushed for it to go open-source.

This statement is more like a waiver you'd like to sign while using some entertainment equipements.