3rd Person movement in relation to camera

I've been trying for a while to try and understand how to move a character in relation to a camera's rotation. At the moment the movement that I have is that W is moving away from the camera, A and D are strafing to the left or right. Moving back simply just backs the character into the camera.

I've been following the 3D Buzz tutorial for the 3rd Person Controller System and it is a great tutorial, but the movement that I described above is something that I currently don't want. I want the character to move and rotate properly in relation to the camera.

the following code are the methods that handle the actual movement and rotation of the player:

 void ProcessMotion()
     {
         // Transform MoveVector to World Space
         MoveVector = transform.TransformDirection(MoveVector);
         
         // Normalize Movevector if Magnitude is > 1
         if (MoveVector.magnitude > 1)
             MoveVector = Vector3.Normalize(MoveVector);
         
         // Multiply MoveVector by MoveSpeed
         MoveVector *= Movespeed;
         
         // Mulitply Movevector by DeltaTime
         MoveVector *= Time.deltaTime;
         
         // Move  the Character in World Space
         TP_Controller.CharacterController.Move(new Vector3(MoveVector.x, MoveVector.y, MoveVector.z));
          
     }
     void SnapAlignCharacterWithCamera()
     {
         if (MoveVector.x != 0 || MoveVector.z != 0)
         {
             transform.rotation = Quaternion.Euler(transform.eulerAngles.x,
                                                   Camera.mainCamera.transform.eulerAngles.y,
                                                   transform.eulerAngles.z);
         }
     }

I somewhat understand what everything does on a high level, but I am sadly not very well versed in things like rotation and Quaternions, and so because of this I am left in the dust because I have no idea how to use them. As far as I can understand, the method SnapAlign...() is used to tell the character that the direction that the camera is facing is forwards, but I don't know where to go from here?

So I guess my question is how do I actually move and rotate a character in relation to the camera, so that forward is moving away from the camera, backwards is running towards it and left and right are moving to the left or to the right of the camera.

[EDIT] I may have described my problem a little weirdly, sorry. When I move my character, the player is rotated so that it is facing forward in relation to the character, which causes the problems i am having.

Below is a crudely drawn diagram. (Sorry, I am truly no artist) The player faces forwards at all times, no matter which input is being applied (left, right, back etc.) I need to get get it so that the character rotates properly, e.g it will rotate left to move left.