Liveness, and the two facts you need when something looks wrong.

database names the host and database this process is actually connected to -- host and name only, never credentials. It is here rather than behind auth because the question it answers ("is prod pointed at the right database?") is one you ask when you cannot get in, and because the answer reveals nothing a port scan would not.

Added after production spent five hours on a stale database with a perfectly healthy /health.