No, there's a 0% chance my implementation outputs 1 as that is only 1 byte long. If you outputted 1023 0s and then a 1 there would be a 2^-1024 chance. That probability is too small to practically happen. The heat death of the universe will happen first.

One could try to guarantee unique outputs with

  sleep 1
  date

but the intended requirement of behave_differently is that it produces different output when run on a computer in exactly the same state. Or equivalently, that the program doesn't depend on the computer state, which is true in the more formal setting of universal Turing machines that start from a fixed state.

If you want something that always behaves differently, why not just concatenating something to whatever the output of the other script would be? I'm familiar with the halting problem, but considering the quick description inside this post I don't see which part of it would forbid this

Also remember that we are considering a bash interpreter which implies we at least have access to $RANDOM. The section behave_differently.sh comes from is only focusing on bash. It does not mention Turing machines.