The story of Alfred "Alfie" Cunningham being named a survivor initially due to mistaken identity was accurate. The first list of survivors had an "A. Cunningham" listed and this was assumed to be Alfie. It was actually a steward, Albert Cunningham, who survived.
In the documentary, Bruce Ismay seems to name the first ship "Olympic" at random, though he actually had a real reason for naming the first of the two ships that name. Ismay's father had planned for a ship named "Olympic" to be built before he had died, and the order for that ship was never fulfilled. Ismay chose "Olympic" as the name for the new class of liners in honor of his father's wish.