Push a live signal when a user is added to a room (#26)

Previously GET /api/rooms/mine was only ever fetched once at app mount,
so a room added mid-session stayed invisible until a full page reload
-- add_member had no way to reach an already-open client at all.

Backend: ConnectionManager and Broadcaster (renamed from RoomBroadcaster)
now support per-user channels alongside the existing per-room ones, so a
signal can reach a user's socket even for a room they haven't joined
(and by definition can't have, until this fires). add_member publishes
a room_added event on the target user's channel.

Frontend: the WebSocket connection is no longer scoped to whichever
room is open -- ChatShellPage now owns one persistent connection for
the whole session (including while no room is open, which is exactly
when this bug showed), and ChatPane joins/leaves rooms on top of it.
A room_added event triggers a room-list refetch with no reload needed.

Verified end-to-end in the browser: a user sitting on the empty room
list saw a newly-added room appear live, then chatted in it normally.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>

backend/app/main.py

@@ -12,7 +12,7 @@ from starlette.middleware.sessions import SessionMiddleware
 
 from app.config import settings
 from app.routers import admin, auth, bots, health, push, rooms, signup, uploads, users, webhooks
-from app.ws.broadcaster import RoomBroadcaster
+from app.ws.broadcaster import Broadcaster
 from app.ws.chat import router as ws_router
 from app.ws.connection_manager import ConnectionManager
 from app.ws.presence import Presence
@@ -68,7 +68,7 @@ def create_app() -> FastAPI:
     app.state.connection_manager = ConnectionManager()
     app.state.redis = Redis.from_url(settings.redis_url, decode_responses=True)
     app.state.presence = Presence(app.state.redis)
-    app.state.broadcaster = RoomBroadcaster(app.state.redis, app.state.connection_manager)
+    app.state.broadcaster = Broadcaster(app.state.redis, app.state.connection_manager)
 
     app.include_router(health.router)
     app.include_router(auth.router)