Yes, there is an angular rotation of the ball about its centre.Yes, angular velocity of the ball about its centre is same as the angular velocity of the ball about the fixed point.Explanation:Let the time period of angular rotation of the ball be T.Therefore, we get Angular velocity of the ball about the fixed point = T2π​After one revolution about the fixed centre is completed, the ball has come back to its original position. In this case, the  point at which the ball meets with the string is again visible after one revolution. This means that it has undertaken one complete rotation about its centre.The ball has taken one complete rotation about its centre. Therefore, we have Angular displacement of the ball = 2πTime period = TSo, angular velocity is again T2π​.Thus, in both the cases, angular velocities are the same.