Step 1: The defining data of a Markov chain has two parts: the rule for where it starts and the rule for how it jumps from one step to the next.
Step 2: Suppose only the transition matrix $P$ is given. Two chains could share the exact same $P$ but start in different states, and then produce entirely different sequences of state probabilities $\pi^{(n)} = \pi^{(0)} P^n$. So $P$ alone cannot fix the chain.
Step 3: Suppose instead only the state space is given, or the process is merely labelled a "stochastic process". Neither carries any numerical probability information, so the chain clearly cannot be reconstructed from that alone.
Step 4: Once both the initial distribution $\pi^{(0)}$ and the transition matrix $P$ are known, every finite dimensional distribution $P(X_0=i_0,\ldots,X_n=i_n) = \pi^{(0)}_{i_0}\prod_{k=0}^{n-1} p_{i_k i_{k+1}}$ is fixed uniquely, so nothing more is needed.
\[\boxed{\text{Transition probability matrix with initial distribution}}\]