At 16:01, the original Max traveled to the future in the second time machine. When he arrived, he found there the second Max waiting for him with the first time machine. He moved to the other machine and traveled back to the past, arriving at 15:59. After changing machines again, he traveled to the future in the original time machine. He never returned.
Only these two scenarios are possible. Any other you may try to build would be inconsistent (try and you will see).
It is easy to see that the world line for the machine is the same in both scenarios. What changes is the traveler’s world line.
1. In scenario 1, the traveler does not change machines either in the present or in the future.
2. In scenario 2, the traveler changes machines both in the present and in the future.
Therefore, if time travel to the past were possible, the traveler would not be free. What he does in the present logically determines what he must do in the future. If he changes machines now, he must change in the future. If he doesn’t, he mustn’t.
Nobody doubts that a human being is free to take an elementary decision such as changing chairs or not.
Therefore time travel towards the past must be impossible.