feat(backend): per-tier, per-kind active-game limits with a guest funnel

Cap a player's simultaneous unfinished games per kind (vs_ai, random,
friends) with independent guest and durable-account tiers, held in a new
single-row backend.config table (-1 = unlimited) behind an in-memory cache
and editable live in the admin console (/_gm/limits). Each game is tagged
with games.game_kind on creation.

This replaces the earlier flat MaxActiveQuickGames=10 combined cap: the
per-tier/kind config is the single mechanism, enforced at the same handler
gate (ensureUnderGameLimit by kind on lobby/enqueue) plus the durable
friends cap in CreateInvitation. game.Service.AtGameLimit only resolves the
tier and counts; the limit policy stays at the request edge.

Guests are now refused friend requests, friend-code redemption,
befriend-in-game and invitation creation outright (403 guest_forbidden) --
previously only the UI hid these.

Admin: a kind column in both game lists and the config editor.

Defaults: guest 1 vs_ai / 1 random / 0 friends; durable 10 / 10 / 10.

backend/internal/lobby/lobby.go

@@ -15,17 +15,22 @@ import (
 
 	"scrabble/backend/internal/engine"
 	"scrabble/backend/internal/game"
+	"scrabble/backend/internal/gamelimits"
 	"scrabble/backend/internal/notify"
 )
 
 // GameCreator is the slice of the game domain the lobby needs: starting a seated
-// game and reading a player's initial view of it. game.Service satisfies it.
+// game, reading a player's initial view of it, and testing a caller's active-game
+// cap. game.Service satisfies it.
 type GameCreator interface {
 	Create(ctx context.Context, params game.CreateParams) (game.Game, error)
 	// InitialState returns a seated player's full initial view of a started game, used
 	// to enrich the game_started event so the client renders the new game without a
 	// follow-up fetch.
 	InitialState(ctx context.Context, gameID, accountID uuid.UUID) (notify.PlayerState, error)
+	// AtGameLimit reports whether accountID has reached its tier's active-game cap for kind. The
+	// invitation path uses it to enforce the friends limit before opening one.
+	AtGameLimit(ctx context.Context, accountID uuid.UUID, kind gamelimits.Kind) (bool, error)
 }
 
 // RobotProvider supplies a robot account to substitute for a missing human in
@@ -62,6 +67,9 @@ var (
 	// ErrInvalidInvitation is returned for a malformed invitation (bad player
 	// count, duplicate or self invitee, or unacceptable settings).
 	ErrInvalidInvitation = errors.New("lobby: invalid invitation")
+	// ErrGuestForbidden is returned when a guest attempts a durable-only action (invite a
+	// friend); the friend flow is gated to durable accounts.
+	ErrGuestForbidden = errors.New("lobby: guests cannot invite")
 	// ErrInvitationBlocked is returned when a block stands between the inviter and
 	// an invitee.
 	ErrInvitationBlocked = errors.New("lobby: invitation blocked between accounts")