I think I'll stick with the premise of my claim - but will expand for better accuracy.
It is an overstatement to talk about "occupation" of Japan and Cuba there is no doubt in those two countries the US Navy had a major impact in exposing them to the culture of the US. We still have bases in both places. For 100 years until the 1950's US Military and related personnel were usually the primary exposure to the US culture around the world. After the Civil War - where they went baseball went with them. The roots of the game in Japan in Cuba are based on students in those country's being exposed to the game and having it coupled with US Military or Former Military personnel on the ground interacting with locals to create momentum.
From the wiki pages attached:
Baseball was introduced to Cuba in the 1860s by Cuban students returning from colleges in the United States and American sailors who ported in the country. The sport spread quickly across the island nation after its introduction, with student Nemesio Guillot receiving popular credit date for the game's growth in the mid-19th century.
Baseball was first introduced to Japan as a school sport in 1872 by American Horace Wilson,[3] an English professor at the Kaisei Academy in Tokyo. The first organized adult baseball team, called the Shimbashi Athletic Club, was established in 1878.
Horace Wilson was born in Gorham, Maine.[1] A veteran of the U.S. Civil War, where he fought for the 12th Maine Volunteer Infantry Regiment against the Confederates in Louisiana.
After the war, he was hired by the Japanese government as a foreign adviser to assist in the modernization of the Japanese education system after the Meiji Restoration. He served as a professor of English at Kaisei Gakko, the forerunner of Tokyo Imperial University.[2