how to make location names appear in higher scales?

The map renderings available at OpenStreetMap.org use pre-rendered raster images. That is, the server uses pre-determined rules to decide what to display at each zoom level and then generates static image files that are shown to you. Since these are static images, there isn't a way to show a different thing for each user, or have fancy actions like mouseover for map features. There's work under way to make a dynamic rendering (similar to the way Google Maps works), but that's still a work in progress.

For the islands in your first example, the information regarding each island is currently mapped as a single node. The map rendering rules have apparently been set up to only show those starting at z16. Since a node doesn't give any clue to the rendering server of how big a given island is, the map maintainers had to choose a zoom level to use as a default. A better way to map these would be to have the name and place=island tags on the actual island outline, and then the map rendering would be able to display the name at appropriate zoom levels based on the size of the island. If you look at this area where the islands are mapped in this way, you can see that the larger islands have their names visible much earlier than the smaller ones.

With the village example, the map maintainers have decided to only start showing names for villages starting at z12. They've likely chosen this as a compromise that they've found works best worldwide. If they have things showing up too early, the map could become very cluttered.