How to make full local export of map part with standart OSM layer?

We have a rather strange problem with OSM.

We need to install an interactive map for a specific area on our site, but we lack the usual export module, which is accessible by a separate key inside the OSM site.

We would like to find out how we can copy a certain area of the map by its coordinates, and so that in addition we can use all its features, such as a beautiful renderer, and not a lot of objects on a white background, notes on the map, as well as the possibility of using the OSMR routing service.

At the moment, after exporting, we only get the designations on the map, without a schematic background, which is presented on the official website as a standard layer.