Thank you a lot Nina!
I just managed to make it work following your steps and using the Debug.Log.
I also have discovered that Normalize() does modify the vector normalizing it, just like ToString() that turns an object into a string.
Here is my solution hoping it will help someone in the future:
private void Move()
{
Vector2 movement = new Vector2(Input.GetAxisRaw("Horizontal"), Input.GetAxisRaw("Vertical"));
movement.Normalize();
movement *= Time.deltaTime * moveSpeed;
float playerPosX = transform.position.x + movement.x;
float playerPosY = transform.position.y + movement.y;
transform.position = new Vector2(Mathf.Clamp(playerPosX, xMin, xMax), Mathf.Clamp(playerPosY, yMin, yMax));
//transform.position = (Vector2)transform.position + movement;
//transform.position = new Vector2(Mathf.Clamp(transform.position.x + movement.x, xMin, xMax), Mathf.Clamp(transform.position.y + movement.y, yMin, yMax));
//Debug.Log(transform.position.x + " " + transform.position.y);
}