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/components/recording-card/recording-card.svelte

@@ -8,10 +8,12 @@
   interface Props {
     recording: Recording;
     onPlay?: (id: string) => void;
+    onPublish?: (id: string) => void;
+    onUnpublish?: (id: string) => void;
     onDelete?: (id: string) => void;
   }
 
-  let { recording, onPlay, onDelete }: Props = $props();
+  let { recording, onPlay, onPublish, onUnpublish, onDelete }: Props = $props();
 
   function formatDuration(ms: number): string {
     const totalSeconds = Math.floor(ms / 1000);
@@ -149,12 +151,35 @@
           {$_("recording_card.play")}
         </Button>
       {/if}
+      {#if onPublish && recording.status === "draft"}
+        <Button
+          size="sm"
+          variant="outline"
+          onclick={() => onPublish?.(recording.id)}
+          class="cursor-pointer border-primary/20 hover:bg-primary/10 hover:text-primary"
+          title={$_("recording_card.publish")}
+        >
+          <span class="icon-[ri--send-plane-line] w-4 h-4"></span>
+        </Button>
+      {/if}
+      {#if onUnpublish && recording.status === "published"}
+        <Button
+          size="sm"
+          variant="outline"
+          onclick={() => onUnpublish?.(recording.id)}
+          class="cursor-pointer border-muted-foreground/20 hover:bg-muted/50 hover:text-muted-foreground"
+          title={$_("recording_card.unpublish")}
+        >
+          <span class="icon-[ri--arrow-go-back-line] w-4 h-4"></span>
+        </Button>
+      {/if}
       {#if onDelete}
         <Button
           size="sm"
           variant="outline"
           onclick={() => onDelete?.(recording.id)}
           class="cursor-pointer border-destructive/20 hover:bg-destructive/10 hover:text-destructive"
+          title={$_("common.delete")}
         >
           <span class="icon-[ri--delete-bin-line] w-4 h-4"></span>
         </Button>