Automatic 3rd person camera

I am trying to make a 'tank control style game and am working on the camera. I would like for it to be automatic like in older games.

Here is what I am working with:

 using System.Collections;
 using System.Collections.Generic;
 using UnityEngine;
 using UnityEngine.InputSystem;
 
 public class Main_Camera_Controller : MonoBehaviour
 {
     public GameObject Camera;
     public Vector3 _offset;
     public Vector3 rawInputMovement;
     public float speed;
     [Range(0.01f, 1.0f)]
     public float SmoothFactor;
 
     private void Start()
     {
         _offset = Camera.transform.position - this.transform.position;
     }
    void LateUpdate()
     {
 
         if (rawInputMovement.x < 0)
         {
             Quaternion camTurnAngle = Quaternion.AngleAxis(transform.rotation.y * speed * Time.deltaTime, Vector3.up);
             _offset = camTurnAngle * _offset;
 
         }
         else if (rawInputMovement.x > 0)
         {
             Quaternion camTurnAngle = Quaternion.AngleAxis(-transform.rotation.y * speed * Time.deltaTime, Vector3.down);
             _offset = camTurnAngle * _offset;
         }
         else
         {
             // get camera to be centered on the player again somehow...
 
         }
         Vector3 newPosHorizontal = transform.position + _offset;
         Camera.transform.position = Vector3.Lerp(Camera.transform.position, newPosHorizontal, SmoothFactor/Time.deltaTime);
         Camera.transform.LookAt(this.transform);
     }
     public void onMovement(InputAction.CallbackContext value)
     {
         Vector2 inputMovement = value.ReadValue<Vector2>();
         rawInputMovement = new Vector3(inputMovement.x, 0, inputMovement.y);
     }
 
 }

I am not sure as to how to get the camera to move back behind the player when moving forward.

The way I am doing it is dependant on the player's rotation at the moment and probably not the best approach? I originally had it set by using the input values from a gamepad but this seemed to be more akin to what I am aiming for.

I am not sure; this is my first dive into how cameras work. Any info would be appreciated.