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_test.go

@@ -1,6 +1,7 @@
 package server
 
 import (
+	"context"
 	"net/http"
 	"testing"
 	"time"
@@ -111,3 +112,35 @@ func TestMoveRecordDTOFrom(t *testing.T) {
 		t.Fatalf("move dto mismatch: %+v", dto)
 	}
 }
+
+// TestGameDTOCarriesSeatSnapshot checks the DTO carries each seat's display-name
+// snapshot (so a disguised robot's per-game name reaches the REST views, not just live
+// events) rather than leaving it for an account lookup.
+func TestGameDTOCarriesSeatSnapshot(t *testing.T) {
+	g := game.Game{
+		Variant: engine.VariantEnglish, Players: 2, TurnTimeout: time.Hour,
+		Seats: []game.Seat{
+			{Seat: 0, AccountID: uuid.New(), DisplayName: "Звёздный_Барс2"},
+			{Seat: 1, AccountID: uuid.New(), DisplayName: "Тест"},
+		},
+	}
+	dto := gameDTOFromGame(g)
+	if dto.Seats[0].DisplayName != "Звёздный_Барс2" || dto.Seats[1].DisplayName != "Тест" {
+		t.Fatalf("seat snapshot not carried into DTO: %+v", dto.Seats)
+	}
+}
+
+// TestFillSeatNamesKeepsSnapshot checks fillSeatNames leaves a seat whose snapshot is
+// already set untouched (no account lookup), so the per-game name is not overwritten by
+// the robot's seeded account name.
+func TestFillSeatNamesKeepsSnapshot(t *testing.T) {
+	s := &Server{} // accounts is unused: a present snapshot must skip the lookup
+	dto := &gameDTO{Seats: []seatDTO{
+		{Seat: 0, AccountID: uuid.New().String(), DisplayName: "Звёздный_Барс2"},
+		{Seat: 1, AccountID: uuid.New().String(), DisplayName: "Тест"},
+	}}
+	s.fillSeatNames(context.Background(), dto, map[string]string{})
+	if dto.Seats[0].DisplayName != "Звёздный_Барс2" || dto.Seats[1].DisplayName != "Тест" {
+		t.Fatalf("fillSeatNames overwrote a seat snapshot: %+v", dto.Seats)
+	}
+}