Hello Stefan,
The osm_domains.xml file should get installed with the ArcGIS Editor for OSM. It sounds like perhaps your instance is not finding it at the location where it wants to look for it. Here is how to check: In ArcMap, enable the Editor toolbar (e.g., Customize menu at top, select "Toolbars", make sure "Editor" is checked). Then when you click on the "Editor" button on the Editor toolbar, you'll see at the bottom of the menu "Options". Click Options, and select the OpenStreetMap tab in the Options window. The second field in that window is "OSM Domains", and should be a filepath to the osm_domains.xml file. Check the following:
1) Does the domain listed there exist on your computer?
2) If so, is there an osm_domains.xml file at that location (probably not)?
3) In either case, do this: Browse to the place where you installed ArcGIS Editor for OpenStreetMap (e.g., typically something like C:\Program Files\ESRI\OSMEditor) and open the "Bin" folder. Is there an "osm_domains.xml" file there? If so, update this OMS Domains parameter in the OpenStreetMap options on the Editor toolbar to the location of your osm_domains.xml file - something like C:\Program Files\ESRI\OSMEditor\Bin\osm_domains.xml. Then Click OK. ArcMap should remember this preference and you should no longer see the error.
Let me know if this works for you.
Christine