the expected output for "core_update": was "true" because there ist a new version avalable (my understanding).
Not inside containers.
Inside containers, the default behavior is to only show updates for the image (as you can see on the web interface footer). Individual component versions will never show "update: true" inside containers.
What is the reason? The docker-installation has no separate value for a new version. Is it possible that the "api" delivers the same values in a docker-installation or an additional value for a new docker-version?
Checking the repo makes no sense for a Pi-hole container.
A dockered Pi-hole is not to be updated via pihole -up, but rather by downloading a new image.
Furthermore, the Docker image has a separate release cycle.
It follows Pi-hole repo releases with a bit of a lag, and Docker specific issues may prompt a new image without requiring any changes to the regular Pi-hole repo.
Edit:
If you want to determine programatically whether an update is available for a container, you probably could consider to check Docker tags instead.