Yes, I do think that at least one of the Anastasia impersonators actually believed she was Anastasia . . . as the adage goes, "Thinking makes it so"! (Just as I believe a certain famous United States sports figure convinced himself that he didn't kill his wife, even though all rational evidence points to him being the murderer . . . but I digress . . . :-/ )
Strange as it seems to us now, Anastasia was actually the most obscure of the four girls at the time of the abdication. Even Marie had a coming-out party of sorts . . . this was the occasion, alas, where she slipped and fell down the steps! But by 1914 she was becoming fairly well recognized, due in great part to her physical maturity implying she was of marriageable age.
Yes, by the time Anastasia reached 17, her features would have been those of her adult self . . . but formal, official photographs of the family were not as plentiful by the beginning of the war, when she had just turned 14. I don't know that the women who claimed to be Anastasia were consciously choosing to be Anastasia, but anyone who wanted to financially back one of these ladies, or who wanted to believe in the survival of a royal family member, probably would find it easier to reconcil to the survival of Anastasia, for the reasons I mentioned above.