feat: honest AI opponent in quick game

New Game's quick game gains an explicit opponent selector — 🤖 AI (default)
or 👤 Random player. AI starts a game seated with a pooled robot that joins
and moves at once: no per-move timeout (a 7-day inactivity loss reusing the
turn-timeout sweeper), chat/nudge disabled, no statistics, the opponent shown
as 🤖 everywhere. The random path (disguised robot) is unchanged.

Driven by one game flag (games.vs_ai), set only on AI-started games so the
disguised path is never revealed; Matchmaker.StartVsAI seats the robot
directly (no open pool); the robot replies event-driven via the game service's
after-commit/after-create hook. Wire: vs_ai on EnqueueRequest + GameView.

backend/internal/server/dto.go

@@ -94,8 +94,11 @@ type gameDTO struct {
 	ToMove               int    `json:"to_move"`
 	TurnTimeoutSecs      int    `json:"turn_timeout_secs"`
 	MultipleWordsPerTurn bool   `json:"multiple_words_per_turn"`
-	MoveCount            int    `json:"move_count"`
-	EndReason            string `json:"end_reason"`
+	// VsAI marks an honest-AI game: the opponent is shown as 🤖 and chat/nudge/add-friend
+	// are suppressed in the client.
+	VsAI      bool   `json:"vs_ai"`
+	MoveCount int    `json:"move_count"`
+	EndReason string `json:"end_reason"`
 	// LastActivityUnix is the lobby sort key: the current turn's start for an active
 	// game, the finish time once finished.
 	LastActivityUnix int64     `json:"last_activity_unix"`
@@ -220,6 +223,7 @@ func gameDTOFromGame(g game.Game) gameDTO {
 		ToMove:               g.ToMove,
 		TurnTimeoutSecs:      int(g.TurnTimeout.Seconds()),
 		MultipleWordsPerTurn: g.MultipleWordsPerTurn,
+		VsAI:                 g.VsAI,
 		MoveCount:            g.MoveCount,
 		EndReason:            g.EndReason,
 		LastActivityUnix:     last.Unix(),