fix(robot): show the per-game name in REST game views, not the account name

The seat display-name snapshot only reached the live-event path (seatNames);
the REST DTO layer still resolved seat names from the account store
(fillSeatNames), so the game screen and lobby list showed the robot's seeded
account name ("Женя") while the your_turn toast showed its per-game name
("Звёздный_Барс2"). Carry the snapshot into gameDTOFromGame and have
fillSeatNames fall back to the account only for a seat with no snapshot (a
pre-snapshot legacy row). Friends and invitations keep account names (the
persistent identity, not a per-game disguise).

backend/internal/server/dto.go

@@ -77,8 +77,9 @@ type moveRecordDTO struct {
 	Total   int       `json:"total"`
 }
 
-// seatDTO is one seat's public standing. DisplayName is resolved from the account
-// store by the handler (the game domain keys seats by account id only).
+// seatDTO is one seat's public standing. DisplayName is the seat's display-name snapshot
+// (the per-game name, frozen when the seat was taken); fillSeatNames fills it from the
+// account store only when a seat has no snapshot yet (a pre-snapshot legacy row).
 type seatDTO struct {
 	Seat        int    `json:"seat"`
 	AccountID   string `json:"account_id"`
@@ -199,19 +200,21 @@ const awayTimeLayout = "15:04"
 func gameDTOFromGame(g game.Game) gameDTO {
 	seats := make([]seatDTO, 0, len(g.Seats))
 	for _, s := range g.Seats {
-		// An open game's still-empty opponent seat has no account: emit an empty id (the
-		// display name is left empty by fillSeatNames) so the client shows "searching for
-		// opponent" rather than the nil-UUID.
+		// An open game's still-empty opponent seat has no account: emit an empty id (and an
+		// empty display name) so the client shows "searching for opponent" rather than the
+		// nil-UUID. DisplayName carries the seat's per-game snapshot; fillSeatNames fills
+		// only the seats still without one from the account store.
 		accountID := ""
 		if s.AccountID != uuid.Nil {
 			accountID = s.AccountID.String()
 		}
 		seats = append(seats, seatDTO{
-			Seat:      s.Seat,
-			AccountID: accountID,
-			Score:     s.Score,
-			HintsUsed: s.HintsUsed,
-			IsWinner:  s.IsWinner,
+			Seat:        s.Seat,
+			AccountID:   accountID,
+			DisplayName: s.DisplayName,
+			Score:       s.Score,
+			HintsUsed:   s.HintsUsed,
+			IsWinner:    s.IsWinner,
 		})
 	}
 	last := g.TurnStartedAt