Thank you, Wayne, for the hint 😀! No, I didn’t know that new option.
So, do I understand it correctly if I think it like this:
If you parent an object “A” simply with the “Object” option, Blender kind of takes a snapshot of the global coordinate system for “A”. Now, you rotate the parent object “B” which also rotates this “snapshot” coordinate system which is the local coordinate system for the child object “A”. That’s why the coordinates of the child object still look the same as they were just one step before partenting “A” to “B”.
Now “Object (Keep Transform Without Inverse)”: It simply copies the local coordinate system of the parent object. Am I right?
And “Apply Parent Inverse” can convert your child’s local coordinate system created with the “Object” parenting option into a copy of the parent’s local coordinate system so that you get the same as directly parenting with “Object (Keep Transform Without Inverse)”. Correct?