Audrey Walsworth was at her home in Naples, Fla., watching the “Today” show in the spring of 2017 when a young woman named Cassie De Pecol appeared on screen. “Meet the first woman to travel to every country in the world,” the chyron read. Ms. Walsworth, 87, was shocked. She watched as a blond woman in her late 20s detailed her exploits and proclaimed herself, “the first woman on record to travel to every country in the world.”