No, they didn't - the 'mark' simply identified that character as being a character transferred from version 1.0 - all 1.0 characters got that mark, and it had nothing to do with Legacy status or how far you got in the storyline (the later of which was the deciding factor whether or not you got the 'special' storyline with the opening of the character 'warping' in and Minfilia remembering who you were, and the special extra bit of dialogue in the ending where everyone finally remembers who you are and everything you did in 1.0.).

Unfortunately SE made this whole thing rather confusing, using the term 'Legacy' loosely to describe contradictory things. But ah well.