Yes, that is generally true in a way.
However, we use 'in' with the big islands only if they have another geographical name. For example, he lives ON the island of Manhattan, but IN New York City. There are weather instruments ON the island of Greenland, but IN Greenland.
If we refer to the city of Miami Beach, (it is a city), we say that he lives IN Miami Beach. But if we refer to just the island, we say ON the Beach.
But, there are exceptions. We say that they live ON Long Island or ON Whidbey Island, both large, inhabited islands.
The use of the preposition may vary from place to place, but in general, of the islands that I know and know of, they are called by the names I have described.
It would be interesting if our members would add islands, island cities, island countries to this list. Let's see what we come up with.