The world of Taekwondo is a convoluted mess, so I can understand your confusion. I do not claim to understand it all myself but here is a quick synopsis.
Korea originally had the ancient martial art of Taekkyon that eventually died out leaving no written record, so no one knows exactly what it entailed (was it an martial art, a folk dance, a sport, or a game?). During the Japanese occupation of Korea during the first half of the 1900s, many young Koreans went to, or were taken to, Japan, where they studied karate. After World War II and the emancipation of Korean from Japanese rule in 1945, some Koreans came back to Korea where they taught the karate they had learned in Japan. Over the years, the karate they taught became influenced by Korean traditions and remnants of what people thought was Taekkyon.
In 1955, the predominate masters met to decide what to call their new art. General Choi suggested the name Taekwondo and the name was adopted. Choi formed the ITF to govern Taekwondo and was its first president. Choi later claimed to have founded Taekwondo, not just name it (some dispute that he even initially proposed the name), which upset the other masters. Choi was born in what became North Korea (the sworn enemy of South Korea) after the Korean War, so he had connections and sympathies with North Korea, which greatly upset the South Koreans to the point that Choi and the ITF were forced out the country in the 1960’s. To this day, South Koreans believe there is a direct connection between the ITF and North Korea.