Change forward control on mouse rotation

Hello there,

I'm currently developing a character controller. I have a camera that smooths follow the player, and a script to control that player, using a character controller.

I've added that when rotating the mouse, player face always look at the pointer. In what I have, "W" is always "Camera.Forward" but now I have some trouble telling the script that "W" is always where the face of the player is looking, and "A" "D" and "S" are relative.

I'm attaching the character code and some screen-shots. Any help is welcome.

     void Start () 
     {
         charController = GetComponent <CharacterController>();
         vertSpeed = minFall;
         camera = Camera.main;
     }
     
     void Update () 
     {
         Vector3 movement = Vector3.zero;
         
         float horInput = Input.GetAxis("Horizontal");
         float vertInput = Input.GetAxis ("Vertical");
         if (horInput !=0 || vertInput !=0)
         {
            
            if(Input.GetKey(KeyCode.LeftShift)) 
             {
             movement.x = horInput * runSpeed;
             movement.z = vertInput * runSpeed;
             movement = Vector3.ClampMagnitude(movement, runSpeed);
             } 
             else
             {
             movement.x = horInput * moveSpeed;
             movement.z = vertInput * moveSpeed;
             movement = Vector3.ClampMagnitude(movement, moveSpeed);
             }
             Quaternion tmp = target.rotation;
             target.eulerAngles = new Vector3 (0, target.eulerAngles.y, 0);
             movement = target.TransformDirection (movement);
             target.rotation = tmp;
             
             Quaternion direction = Quaternion.LookRotation(movement);
             transform.rotation = Quaternion.Lerp(transform.rotation, direction, rotSpeed * Time.deltaTime);
         }
         
         movement.y = vertSpeed;
         movement *= Time.deltaTime;
         charController.Move(movement);
 
         FixedControlMouse();
     }
 
       void FixedControlMouse ()
     {
         Ray ray = camera.ScreenPointToRay (Input.mousePosition);
         Plane groundPlane = new Plane (Vector3.up, Vector3.zero);
         float rayDistance;
 
         if (groundPlane.Raycast(ray, out rayDistance))
         {
             Vector3 point = ray.GetPoint(rayDistance);
             Debug.DrawLine(ray.origin, point, Color.green);
             LookAt(point);
         }
     }
 
     void LookAt (Vector3 lookPoint)
     {
         Vector3 hcPoint = new Vector3(lookPoint.x, transform.position.y, lookPoint.z);
         transform.LookAt(hcPoint);
     }

I'm thinking about some "transform.forward" but it's not working.