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The details always remain in the map. They only change in the images displayed to the user. Those images depend on the tools used to convert the map to images and how they are configured. openstreetmap.org provides four different examples, but there are more.

Also, the openstreetmap servers are not intended for production use by end user applications. For that you should maintain an offline copy of the relevant part of the map database and use that as your source.

Both of these lead to the conclusion that you need to construct your own map tile server (I’m assuming you are using tiled slippy maps, and not applications that display directly from the vector data) and then configure it to implement the policies you have chosen for what gets displayed at what zoom level, or at all.

There are commercial operations that will provide custom slippy map tiles for you, at a price.