It seems that similar questions have come up a few times regarding this, but I'm struggling to understand the answers.
My question is a bit more basic, can the difference between the strong markov property and the ordinary markov property be intuited by saying:
"the markov property implies that a markov chain restarts after every iteration of the transition matrix. By contrast, the strong markov property just says that the markov chain restarts after a certain number of iterations given by a hitting time T"?
Moreover, would this imply that with a normal markov property a single transition matrix will be enough to specify the chain, whereas if we only have the strong property we may need T different transition matrices?