Third-person camera snaps to 270 degrees on activation

Your issue is the line:

 cameraRotation= Quaternion.Euler(mouseY, mouseX, 0f);

This orientation probably looks at the side of your character. This rotation is global and is not relative to the position of character/player.

Try modifying it to take the last known rotation of the camera like this:

     public Collider target;
     // The player's gameobject - we want to look at this
     new public Camera camera;
     // The main camera - the script should move and rotate this
     public float rotationUpdateSpeed = 60.0f,
     verticalRotationSpeed = 20.0f,
     followUpdateSpeed = 10.0f;
     // Variables to fine-tune camera performance
     private float optimalDistance;
     //How far the camera should be from the player
     private float mouseX;
     private float mouseY;
     //Store the GetAxis of the mouse in order to aim the camera 
 
     private Vector3 lastCameraRotationEuler;
 
     Quaternion cameraRotation;
     Vector3 cameraPosition;
     
     void Start (){
         optimalDistance = (camera.transform.position - target.transform.position).magnitude;
     }
     
     
     void Update (){
         if (Input.GetMouseButtonUp (1)) {
             mouseX = target.transform.rotation.z;
             mouseY = target.transform.rotation.y;
         }
     }
     
     
     void LateUpdate ()
     {
         if (Input.GetMouseButton (1)) {
             FreeUpdate ();
         } 
         else {
             FollowUpdate();
         }
         //Screen.lockCursor = true;
         
     }
     
     
     void FollowUpdate (){
         // Camera follows behind the player
         Vector3 cameraForward = target.transform.position - camera.transform.position;
         cameraForward = new Vector3 (cameraForward.x, 0.0f, cameraForward.z);
         float rotationAmount = Vector3.Angle (cameraForward, target.transform.forward);
         rotationAmount *= followUpdateSpeed * Time.deltaTime;
         
         if (Vector3.Angle (cameraForward, target.transform.right) < Vector3.Angle (cameraForward, target.transform.right * -1.0f)){
             // Rotate to the left if the camera is to the right of target forward
             rotationAmount *= -1.0f;
         }
         
         
         camera.transform.RotateAround (target.transform.position, Vector3.up, rotationAmount);
         lastCameraRotationEuler = camera.transform.rotation.eulerAngles;
     }
     
     void FreeUpdate (){
         //Camera controlled by mouse
         mouseX += Input.GetAxis("Mouse X") * rotationUpdateSpeed * 0.02f;
         mouseY -= Input.GetAxis("Mouse Y") * verticalRotationSpeed * 0.02f;
         
         cameraRotation= Quaternion.Euler(lastCameraRotationEuler.x+ mouseY, lastCameraRotationEuler.y+mouseX, lastCameraRotationEuler.z);
         cameraPosition= cameraRotation* new Vector3(0.0f, 2.0f, -optimalDistance) + target.transform.position;
         
         camera.transform.rotation = cameraRotation;
         camera.transform.position = cameraPosition;
         
     }

Just three extra lines to add using another Vector3 that stores the last camera position from the FollowUpdate() method. Just do a find for lastCameraRotationEuler in the code if you are unsure which lines are new.