That interface is the "osm website" or usually called "osm.org", a related project, but it has a different license and maintainers, and it's just only one way to see the OSM databse. Source code is here: https://github.com/openstreetmap/openstreetmap-website

If you click on the Edit button by default you will be redirected to another project called iD editor, again with different license and maintainers: https://github.com/openstreetmap/iD

I usually use JOSM, Overpass Turbo and OsmCha to see the live database, and they are all semi-independent projects.