Hello, can I edit my only for me, not to website?
Welcome to this forum, Milosz! What you require is basically contrary to the primary intention and mission of openstreetmap: the goal is to create a database of geographical information for all. Creating a map from that database is a second matter, the default map is just one example of that. Read more about it at https://wiki.openstreetmap.org/wiki/Standard_tile_layer . But there are many ways of consulting the database, and it is quite possible that the umap thing mentioned does what you are looking for; there may well be others, too.