Under federal law to be called Bourbon it has to be distilled in the United States, not just Bourbon County or even Kentucky.
https://www.law.cornell.edu/cfr/text/27/5.22
You are correct, and to sell as bourbon in the United States, it must conform to this statue. However, Japan, China, or any other country, can call their hooch bourbon but can only sell it outside the U.S. Be very careful buying and drinking alcoholic beverages outside the U.S. Antifreeze is the most popular additive to make cheap wine taste like the good stuff. Most room bars in the Carib are stocked with adulterated booze.