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/handlers_game.go

@@ -66,10 +66,15 @@ type complaintRequest struct {
 	Note string `json:"note"`
 }
 
-// fillSeatNames resolves each seat's display name from the account store, memoising
-// across seats and games within one request.
+// fillSeatNames fills each seat's display name from the account store — memoising across
+// seats and games within one request — for the seats that carry no per-game snapshot yet
+// (a pre-snapshot legacy row). A seat whose snapshot is already set is left untouched, so
+// a disguised robot keeps its per-game name rather than reverting to its account name.
 func (s *Server) fillSeatNames(ctx context.Context, g *gameDTO, memo map[string]string) {
 	for i := range g.Seats {
+		if g.Seats[i].DisplayName != "" {
+			continue // the seat already carries its per-game snapshot
+		}
 		id := g.Seats[i].AccountID
 		name, ok := memo[id]
 		if !ok {