Sampling Multiple Times From a Markov Chain
One correct way is to repeat the following 10 times: pick a random starting state, and apply the Markov chain for $T$ steps.

I don't know whether your first proposal gives the correct distribution in all cases.  You can consider the Markov chain $A \leftarrow B \rightarrow C$ with self-loops $A \to A$ and $C \to C$ and more complicated examples in that vein.

Your second proposal doesn't give the correct distribution.  The $T$th and $T+1$th state are highly correlated: for instance, they will always be adjacent -- something that wouldn't be true of 10 independent samples from the steady state distribution.