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/encode.go

@@ -37,6 +37,7 @@ func toWireGame(g GameSummary) wire.GameView {
 		TurnTimeoutSecs:      g.TurnTimeoutSecs,
 		MultipleWordsPerTurn: g.MultipleWordsPerTurn,
 		VsAI:                 g.VsAI,
+		Kind:                 g.Kind,
 		MoveCount:            g.MoveCount,
 		EndReason:            g.EndReason,
 		Seats:                seats,

backend/internal/notify/notify_test.go

@@ -115,7 +115,7 @@ func TestGameOverPayloadRoundTrips(t *testing.T) {
 func TestOpponentMovedPayloadRoundTrips(t *testing.T) {
 	uid, gid := uuid.New(), uuid.New()
 	move := engine.MoveRecord{Player: 1, Action: engine.ActionPlay, Words: []string{"STOOL"}, Score: 24, Total: 130}
-	summary := notify.GameSummary{ID: gid.String(), MoveCount: 9, ToMove: 0, Seats: []notify.SeatStanding{{Seat: 1, Score: 130}}}
+	summary := notify.GameSummary{ID: gid.String(), MoveCount: 9, ToMove: 0, Kind: 2, Seats: []notify.SeatStanding{{Seat: 1, Score: 130}}}
 	in := notify.OpponentMoved(uid, gid, move, summary, 42)
 	if in.Kind != notify.KindOpponentMoved {
 		t.Fatalf("kind = %q", in.Kind)
@@ -132,7 +132,7 @@ func TestOpponentMovedPayloadRoundTrips(t *testing.T) {
 	if m == nil || m.Player() != 1 || string(m.Action()) != "play" || m.Total() != 130 {
 		t.Fatalf("move wrong: %+v", m)
 	}
-	if g := ev.Game(nil); g == nil || g.MoveCount() != 9 || g.ToMove() != 0 {
+	if g := ev.Game(nil); g == nil || g.MoveCount() != 9 || g.ToMove() != 0 || g.Kind() != 2 {
 		t.Fatalf("game summary wrong: %+v", ev.Game(nil))
 	}
 }

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
 }
 
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