In a world of over-exposure, St. Barthelemy retains its luxe mystique as one of the most exclusive, most chic islands in the Caribbean. Off the beaten track, and with French flair, this tiny island has an outsized reputation and appeal for those in the know. Maybe it’s partly because of its (several) names. The French Caribbean island’s long and formal name is hardly ever used, in favor of its nickname, whispered among the high-end travelers and yachters who gather in this...