Island = the island/isle of + name: ‘The holiday resort island of Langkawi’, ‘The Isle of Skye’. ‘I was on holiday with my parents on the island of Capri.’
When island means ‘a mass of land surrounded by water’, or is part of a name, the usual preposition is on (NOT in): ‘The prisoners were left on a small island with neither food nor drinking water.’
When an island is considered in terms of its people, tradition, and economy etc., both in or on are used: ‘No serious outbreak of cholera has been reported in the island for over twenty years.’