A Markov chain is commonly understood to be a time-discrete Markov process. Intuitively, it’s a “chain” because you can “single out” its states in time rather than intervals. That’s also Markov’s original definition. Instead of Wassermann one can look up the notion in Wikipedia. A path is a notion relevant to the state space of a process - it’s a realization of states at every single point in time for the time span of interest.