fix(games): carry game kind on live events + fix the New Game lock funnel

Address review of the active-game limit lock:

- Live-event GameViews (opponent_moved, match_found, game_started, ...) did
  not carry game_kind, so a lobby patch from an event zeroed a game's kind
  and the client's per-kind count under-counted — a capped kind read as
  free (a disabled start instead of the lock, and a wrong per-kind result).
  Thread Kind through notify.GameSummary → the event GameView.

- New Game screen refreshes the lobby games on mount so the per-kind count
  reflects the current set, not a stale cached snapshot.

- The guest funnel's login button routes to the profile screen (the account
  controls), not settings.

- Copy: the guest prompt is "sign in to use all the game's features"; the
  durable notice is "finish your active games to start a new one".

Regression tests: the notify opponent-moved payload carries kind; the client
lock locks only the reached kind (vs_ai at cap leaves random open).

ui/src/lib/gamelimits.test.ts

@@ -74,4 +74,12 @@ describe('isKindLocked', () => {
   it('a durable account stays well under its higher cap', () => {
     expect(isKindLocked([game(GameKind.random, 'active')], durable, GameKind.random)).toBe(false);
   });
+
+  it('locks only the reached kind: at the vs_ai cap, random with no games stays open', () => {
+    const limits = { vsAi: 3, random: 2, friends: 1 };
+    const games = [game(GameKind.vsAi, 'active'), game(GameKind.vsAi, 'active'), game(GameKind.vsAi, 'active')];
+    expect(isKindLocked(games, limits, GameKind.vsAi)).toBe(true);
+    expect(isKindLocked(games, limits, GameKind.random)).toBe(false);
+    expect(isKindLocked(games, limits, GameKind.friends)).toBe(false);
+  });
 });

ui/src/lib/i18n/en.ts

@@ -380,9 +380,9 @@ export const en = {
   'new.start': 'Start game',
   'new.invited': 'Invitation sent.',
   'new.limitGuestTitle': 'More games?',
-  'new.limitGuestBody': 'Sign in or create an account to play several games at once.',
+  'new.limitGuestBody': 'Sign in or create an account to use all the game features.',
   'new.limitDurableTitle': 'Game limit',
-  'new.limitDurableBody': 'You have reached the limit of simultaneous games — finish a current one first.',
+  'new.limitDurableBody': 'You have reached the limit. Finish your active games to start a new one.',
   'new.limitLogin': 'Sign in',
   'new.noFriends': 'Add friends first to invite them.',
   'new.opponentAI': 'AI',

ui/src/lib/i18n/ru.ts

@@ -380,9 +380,9 @@ export const ru: Record<MessageKey, string> = {
   'new.start': 'Начать игру',
   'new.invited': 'Приглашение отправлено.',
   'new.limitGuestTitle': 'Больше партий?',
-  'new.limitGuestBody': 'Войдите или создайте учётную запись, чтобы играть несколько партий одновременно.',
+  'new.limitGuestBody': 'Войдите или создайте учётную запись, чтобы воспользоваться всеми функциями игры.',
   'new.limitDurableTitle': 'Лимит игр',
-  'new.limitDurableBody': 'Вы достигли лимита одновременных игр, сначала завершите текущие.',
+  'new.limitDurableBody': 'Вы достигли лимита. Доиграйте активные партии, чтобы начать новую.',
   'new.limitLogin': 'Вход',
   'new.noFriends': 'Сначала добавьте друзей, чтобы пригласить их.',
   'new.opponentAI': 'ИИ',