In my theory, the reading ways of kanji letters (=Chinese letters) in Japanese and Chinese are different.
Sound of "Japan" is transformed from Chinese way of reading when ancient time travelled long way from eastern Asia to Europe.
Nippon is the name Japanese call their country: often used as a war cry exactly the same like "USA" calls in sports games.
It should be noteworthy that usually peace-loving Japanese would prefer to call their country "Nihon", because during WW2 they were forced to call their homeland "Dai-Nippon-Teikoku" that means Great Imperial Japan.
"Nippon, cha-cha-cha!" is preferred to cheer girls teams in volleyball games.