fix(chat): no chat field or nudge on a finished game

A finished game is read-only: gate the nudge behind canNudge (active game,
opponent's turn) so it no longer shows on a finished (or otherwise
non-active) game — where the backend rejects it anyway. The message field
is already hidden off your own turn. Extend the finished-game e2e to assert
neither Send nor Nudge is offered.

ui/src/game/ChatScreen.svelte

@@ -26,6 +26,9 @@
   // the turn advances. canSend hides the field once the limit is reached (and always on
   // the opponent's turn).
   const canSend = $derived(canSendChat(isMyTurn, view ? sentThisTurn(messages, myId, view.game.lastActivityUnix) : 0));
+  // The nudge is offered only on the opponent's turn of an active game (you hurry the player
+  // who is to move). A finished game is read-only: neither chat nor nudge is offered.
+  const canNudge = $derived(!!view && view.game.status === 'active' && view.game.toMove !== view.seat);
   // While the auto-match game still has no opponent, chat and nudge are both disabled.
   const waiting = $derived(!!view && view.game.status === 'open');
   const nudgeCooldownSecs = 3600;
@@ -110,4 +113,4 @@
   }
 </script>
 
-<Chat {messages} {myId} {busy} myTurn={isMyTurn} {canSend} {waiting} {nudgeOnCooldown} onsend={sendChat} onnudge={nudge} />
+<Chat {messages} {myId} {busy} myTurn={isMyTurn} {canSend} {canNudge} {waiting} {nudgeOnCooldown} onsend={sendChat} onnudge={nudge} />