More precisely, an initial distribution $m$ of a Markov chain on a state space $X$ determines the associated measure $\mathbf P_m$ on the space of sample paths $X^{\mathbb Z_+}$. The measure $\mathbf P_m$ is shift invariant iff the measure $m$ is stationary. Now, if $m$ is finite (this condition is important; otherwise the following claim is false), then ergodicity of the time shift is equivalent to absence of non-trivial partitions of $X$ into non-communicating subsets.
By the way, your example is really too degenerate: the standard example for difference between ergodicity and mixing for Markov chains is presence of so-called periodic classes $A_1\to A_2\to\dots\to A_k\to A_1$ (the only allowed transitions are from $A_i$ to $A_{i+1}$ mod k). For finite chains this is actually the only reason for difference between ergodicity and mixing, but for general state spaces the situation is more complicated.