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).

backend/internal/notify/payload.go

@@ -35,6 +35,9 @@ type GameSummary struct {
 	EndReason        string
 	Seats            []SeatStanding
 	LastActivityUnix int64
+	// Kind is the game's origin for the active-game limits (0 unknown, 1 vs_ai, 2 random, 3 friends);
+	// it rides live events so a lobby patch keeps the per-kind count correct.
+	Kind int
 }
 
 // AlphabetLetter is one variant alphabet entry (a display-only row) embedded in an