As I remember, the Tickler was the first name she gave and seemed to be more of a trial run than anything. Remember, at the point he told her she had 3 names to give to the Red God, she had no idea he was an assassin.
Once she saw that he killed the Tickler, she started naming Tywin and Joffrey but he said he couldn't guarantee their deaths would be swift. Then she got caught snooping or whatever and needed Tywin's henchman killed immediately.
The "final" name ended up being all the guards on watch the night she escaped. "A girl lacks honor".
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