I think it depends on the character.
"Classic Batgirl" might be Barbara Gordon, but OTOH, "classic Cassandra Cain" would be Cass as Batgirl.
Of course, this gets really complicated with characters like Dick Grayson or Steph Brown (whose heroic identities are pretty much meaningless by now) or with the Flash (since evryone who held the mantle but Bart could be referred to as "classic").