If you intend to 'replace' Bertholys lead knee first move with a trail hip turn toward the ball, be warned that there is a high probability you will come 'over the top'. In most people there is little if any flexibility in reserve to turn the hips without turning the shoulders as well. The knee movement produces an initial lateral movement that delays the hip turn.
If you are not moving your lower body at all until your arms drop then you are doing Dunigan and not Bertholy.