feat: fix recording save and add publish/unpublish support

- Fix broken fetch("/api/sexy/recordings") → use createRecording GraphQL service
- Round duration to integer before sending (GraphQL Int type)
- Add updateRecording mutation to services
- Add publish/unpublish buttons to RecordingCard (draft ↔ published)
- Remove "Go to Play" button from recordings page header
- Add publish/unpublish i18n keys

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

packages/frontend/src/lib/services.ts

@@ -902,6 +902,32 @@ export async function createRecording(
   );
 }
 
+const UPDATE_RECORDING_MUTATION = gql`
+  mutation UpdateRecording($id: String!, $status: String, $public: Boolean) {
+    updateRecording(id: $id, status: $status, public: $public) {
+      id
+      status
+      public
+    }
+  }
+`;
+
+export async function updateRecording(
+  id: string,
+  fields: { status?: string; public?: boolean },
+) {
+  return loggedApiCall(
+    "updateRecording",
+    async () => {
+      const data = await getGraphQLClient().request<{ updateRecording: Recording }>(
+        UPDATE_RECORDING_MUTATION,
+        { id, ...fields },
+      );
+      return data.updateRecording;
+    },
+  );
+}
+
 const DELETE_RECORDING_MUTATION = gql`
   mutation DeleteRecording($id: String!) {
     deleteRecording(id: $id)