How to make a camera move along and arbitrary set left-right axis

So I am trying to make a rail shooter game however I want the camera to always move left or right around the player when the left or right key is pressed. at the moment I am using a rotation on a pivot to determine the camera's position. However, when using the Y-axis and rotating the pivot up, it makes the camera move in a diagonal orbit instead of left to right. How would I be able to set up a system so the camera only moves left and right on the pivot but not diagonal?

Here is my code:

     public GameObject target;
     public float rotateSpeed = 5;
     public float minConstraintsX, maxConstraintsX;
     Vector3 offset;
 
     void Start()
     {
         offset = target.transform.position - transform.position;
     }
 
     void LateUpdate()
     {
         float horizontal = Input.GetAxis("Horizontal") * rotateSpeed;
         float vertical = Input.GetAxis("Vertical") * rotateSpeed;
 
         target.transform.Rotate(0, horizontal, 0, Space.Self);
         target.transform.rotation = ClampRotationAroundXAxis(target.transform.rotation);
         target.transform.Rotate(vertical, 0, 0);
 
         float desiredAngleY = target.transform.eulerAngles.y;
         float desiredAngleX = target.transform.eulerAngles.x;
         Quaternion rotation = Quaternion.Euler(desiredAngleX, desiredAngleY, 0);
         transform.position = target.transform.position - (rotation * offset);
         transform.LookAt(target.transform);
     }
 
     Quaternion ClampRotationAroundXAxis(Quaternion q)
     {
         q.x /= q.w;
         q.y /= q.w;
         q.z /= q.w;
         q.w = 1.0f;
 
         float angleX = 2.0f * Mathf.Rad2Deg * Mathf.Atan(q.x);
 
         angleX = Mathf.Clamp(angleX, minConstraintsX, maxConstraintsX);
 
         q.x = Mathf.Tan(0.5f * Mathf.Deg2Rad * angleX);
 
         return q;
     }

Thanks in advance