openstreetmap.org correctly displays the islands and the relative distances between them!

Sounds like a bug in the app then. Though, OSM apps have their choice on how to interpret the underlying data, so it could be based on a deliberate decision. May be worth reporting if you don't mind revealing your town on their github.

Alternate thought just came to mind - is your map data on OrganicMaps up-to-date?

I hope so. I first heard about it (from this very post on HN), downloaded, and installed the app all within the past 1 hour.

And when I zoomed into my city, it downloaded the map information.

In that case it's as up-to-date as OrganicMaps has to offer, which should be within a couple weeks of OSM's main database.