No, it's really a stochastic process. And if you would like to call it such when explaining it to a novice, be my guest. Then the novice can ask what a stochastic process is and expect to get a full explanation from you.
Again, it needs to be pointed out to you and probably others that harp on irrelevant things, that it is irrelevant that some people, novices probably included, are fuzzy or ignorant of the precise meaning of the technical terms under discussion. That is, that somebody is ignorant of the technical meaning of the term is irrelevant to that fact that stochastic means random, and evolution is a stochastic process, and therefore it is correct to call it a random process.