Going by the letter of the law,
If the death leads to the death of more than the intended, the person will simply die of a heart attack.”
logically implies that the surgeon and pilot will die of heart attacks.
This probably makes the most sense. Were it otherwise, people who know the rules could make themselves (nearly) invincible to the Death Note by having a system by which, if they die, someone else is killed (or kills themself). I say “nearly” because the cause of death could be made to say “die at midnight” for both people, and that might kill them. Obviously, the Death Note user would have to know both people involved, which could be made near-impossible quite easily.
Though, there is the further question of whether “die at midnight” would work. One person's name has to be written first, and at the time at which it is written, it is not allowed. However, writing the other person's name after then makes it allowed again. When are the rules evaluated, and by what means? Death Note has the power to predict the future of its victims, but can it predict the future of its own use? This is into the realm of time paradoxes!
Or, you could just say that they die of a heart attack. ;-)
Edit: For example, if L wanted to make himself invincible to the Death Note, he could have some system by which someone from Wammy's House is killed if he dies. Then, to kill L, a Death Note user needs both L's name and the name and face of that person who gets killed. Even for someone with shinigami eyes, that's virtually impossible to work out.