How to rotate a camera locally around target?

I have seen only examples of camera orbiters where the camera is being rotated on the XZ-plane. But how I can orbit the cam around based from the rotation of the target (Local instead Global Rotation) ?

This is a little visualization I have mashed up from Unity to help you understanding. The camera moves on the sphere, constantly looking at the target (center of the sphere).

Lets assume I want to move my mouse left/right.

On the left side of image:

Normal Behaviour of most cam-orbit scripts, orientating on the green circle parallel to the ground (global Y).

On the right side of image:

This is in what angle the cam should go around. It follows the green circle, which is obviously the Y-Axis of the target.

The same for up/down on the red circle which is representing the targets X-Axis, instead of the global X.

This is my approach so far:

 void LateUpdate () {
     if (!target)
         return;
     x += Input.GetAxis("Mouse X") * xSpeed * distance * Time.deltaTime;
     y -= Input.GetAxis("Mouse Y") * ySpeed * distance * Time.deltaTime;
         
     Quaternion rotation = Quaternion.Euler(y, x, 0) * target.rotation;

     distance = Mathf.Clamp(distance - Input.GetAxis("Mouse ScrollWheel")*distanceMin*3f, distanceMin, distanceMax);
 
     Vector3 position = -Vector3.forward * distance * rotation + target.position;
 
     transform.rotation = rotation;
     transform.position = position;
 }

I hope this is clear enough to understand ;) Thanks in advance!