Yes, exactly. More generally (and for the same reason), any sequence of iid random variables will form a Markov chain. For an example without the Markov proper

Yes, exactly. More generally (and for the same reason), any sequence of iid random variables will form a Markov chain. For an example without the Markov property, consider the sequence of random variables X_1, X_2, ... with X_1 being either -1 or 1 with equal probability, and X_t being normally distributed with mean X_1 and standard deviation 1. Knowing the history X_1, X_2, ..., X_{t-1} gives you the exact distribution of X_t (since you know X_1), while only knowing X_{t-1} gives you much less information. This fails to be a Markov chain because the state X_{t-1} doesn't "remember" which of the two possible distributions is being used.