What makes it "chaotic" isn't just that it'll turn right if you start it leaning a tiny bit right and left if you start it leaning a tiny bit left. What makes it chaotic is that, if you start it in position X, it might make the series of turns RRLRLLLRL... while if you start it in position X+epsilon, it might make the series of turns RRLRLLLLR... or LLRLRRRLR... or some other completely different sequence. It may even make the first hundred turns exactly the same, and then diverge at turn #101.

In other words, any tiny change in the initial conditions, even if it doesn't change the first turn, can change the pattern of turns farther down the line. In chaos, sensitive dependence on initial conditions doesn't just happen at a single point, but throughout the system. It's not just a matter of making or missing the train, but of talking to different people or having slightly different mannerisms because you were 3 minutes and 2.8 seconds early instead of 3 minutes and 2.7 seconds early, and those tiny changes eventually adding up to lead you to Boise instead of Miami.