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/e2e/gamelimit.spec.ts

@@ -28,7 +28,7 @@ test('new game: a start locks at its per-kind cap and the funnel modal opens', a
 
   // Tapping the locked start opens the modal instead of a game, and does not navigate away.
   await start.click();
-  const notice = page.getByText(/reached the limit of simultaneous games/i);
+  const notice = page.getByText(/reached the limit/i);
   await expect(notice).toBeVisible();
   await expect(page).toHaveURL(/\/new$/);
   await page.getByRole('button', { name: /^ok$/i }).click();