Can you tell me how to make a map without transferring all the data from OSM. That is, a clean card to put your data on. For example: the organization and all branches in the country, etc. Just that. (there will be a link below as needed)
There are a variety of companies offering raster or vector “tiles” made from OpenStreetMap data that can then be used as a background map. Some of the providers have a free tier if you have few enough visitors.
switch2osm has some guidance for people wanting to start using pre-made tiles. If you follow either the leaflet or OpenLayers links on that page you will see further information about adding a company’s branches to the map. The examples on that page use the tiles straight from OpenStreetMap.org which is limited to very low volume applications so you might need to change to a different provider if you are expecting more traffic than this.
Just as a bit of extra information, your example site uses raster map tiles from osm.org itself such as https://a.tile.openstreetmap.org/6/48/24.png. The serving tiles side of the Switch2osm site will show you how to create your own raster tiles that you can set up in a Leaflet map along with the “company’s branches” or whatever.
There are lots of other ways of making maps (for example, if you wanted something in an Android app, you’d probably do something different) - but which is best depends on what you want to do.