fix: make leave room button actually remove user from room

When clicking "Leave Room", the user is now properly removed from
room membership (not just marked offline) and navigated to /arcade.

Previously:
- Just navigated to /arcade/rooms
- User remained as offline member of the room
- Socket cleanup only marked user offline

Now:
- Calls POST /api/arcade/rooms/:roomId/leave
- Actually removes user from room_members table
- Navigates to /arcade home page
- Users can be in at most 1 room (or 0 rooms)

🤖 Generated with [Claude Code](https://claude.com/claude-code)

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

apps/web/src/app/arcade/rooms/[roomId]/page.tsx

@@ -196,8 +196,24 @@ export default function RoomDetailPage() {
     }
   }
 
-  const leaveRoom = () => {
-    router.push('/arcade/rooms')
+  const leaveRoom = async () => {
+    try {
+      const response = await fetch(`/api/arcade/rooms/${roomId}/leave`, {
+        method: 'POST',
+        headers: { 'Content-Type': 'application/json' },
+      })
+
+      if (!response.ok) {
+        const errorData = await response.json()
+        throw new Error(errorData.error || `HTTP ${response.status}`)
+      }
+
+      // Navigate to arcade home after successfully leaving
+      router.push('/arcade')
+    } catch (err) {
+      console.error('Failed to leave room:', err)
+      alert('Failed to leave room')
+    }
   }
 
   if (loading) {