feat(chat): add presence metadata and improve web chat core

- add user last_seen_at with alembic migration and persist on realtime disconnect
- extend chat serialization with private online/last_seen, group members/online, channel subscribers
- add Redis batch presence lookup helper
- update web chat list/header to display status counters and last-seen labels
- improve delivery receipt handling using last_delivered/last_read boundaries
- include chat info panel and related API/type updates

web/src/components/ChatList.tsx

@@ -145,7 +145,7 @@ export function ChatList() {
                     </span>
                   )}
                 </div>
-                <p className="truncate text-xs text-slate-400">{chat.type}</p>
+                <p className="truncate text-xs text-slate-400">{chatMetaLabel(chat)}</p>
               </div>
             </div>
           </button>
@@ -283,3 +283,46 @@ function chatLabel(chat: { display_title?: string | null; title: string | null;
   if (chat.type === "group") return "Group";
   return "Channel";
 }
+
+function chatMetaLabel(chat: {
+  type: "private" | "group" | "channel";
+  is_saved?: boolean;
+  counterpart_is_online?: boolean | null;
+  counterpart_last_seen_at?: string | null;
+  members_count?: number | null;
+  online_count?: number | null;
+  subscribers_count?: number | null;
+}): string {
+  if (chat.is_saved) {
+    return "Personal cloud chat";
+  }
+  if (chat.type === "private") {
+    if (chat.counterpart_is_online) {
+      return "online";
+    }
+    if (chat.counterpart_last_seen_at) {
+      return `last seen ${formatLastSeen(chat.counterpart_last_seen_at)}`;
+    }
+    return "offline";
+  }
+  if (chat.type === "group") {
+    const members = chat.members_count ?? 0;
+    const online = chat.online_count ?? 0;
+    return `${members} members, ${online} online`;
+  }
+  const subscribers = chat.subscribers_count ?? chat.members_count ?? 0;
+  return `${subscribers} subscribers`;
+}
+
+function formatLastSeen(value: string): string {
+  const date = new Date(value);
+  if (Number.isNaN(date.getTime())) {
+    return "recently";
+  }
+  return date.toLocaleString(undefined, {
+    day: "2-digit",
+    month: "short",
+    hour: "2-digit",
+    minute: "2-digit"
+  });
+}