Baseball four out rule?

A complicated play indeed, and let me try to explain it to you. There was one out and runners on second and third base. The ball was lined back to the pitcher for the second out and both runners took off as the ball was hit. The pitcher turned and threw the ball to the second baseman and all he had to do was to step on second base before the runner got back and that would have been the third out and no runs would have scored. Instead, the second baseman did not step on the bag and ran toward the base runner coming back to second base and he was tagged out for the third out. However, this means that the runner was in a rundown before he was tagged out and the runner on third base scored before the base runner was tagged out and, therefore, the run counted. All the defensive team had to do was to appeal the play at third base and that runner would have been called out for leaving early and not going back to the bag to tag up, and the run would not have counted. Technically that would have been the fourth out in the inning but it did not happen. Once the defensive team left the field they could not appeal the play and the run counted.