"Markov" is used in the standard memoryless sense. By definition, the graph G represents any distribution p where each variable on the graph is independent of its past given its parents. This is the Markov property.

Ilya is discussing probability distributions p that may or may not be represented by graph G. If every variable in p is independent of its past given its parents in G, then you can use d-separation in G to reason about independences in p.