Ok... After changing some settings, I got this log:
2023-06-13T15:16:56.808411Z ERROR HTTP request{http.method=GET http.scheme="http" http.host=lemmy.pathoris.de http.target=/api/v3/ws otel.kind="server" request_id=6603f30f-4523-4548-a82c-f38c22d1889f http.status_code=101 otel.status_code="OK"}: lemmy_server::api_routes_websocket: only_mods_can_post_in_community: only_mods_can_post_in_community
That explains a lot, I think :-)
Apparently my instance does try to push, as I expected, but the community is protected on lemmy.world. That's a bit surprising still, since I thought it is a test group to test federation and such, but since that wasn't specified anywhere, it's fair enough that this could be a test-group by lemmy.world for lemmy.world :-)