Maggie to friends and family in her youth. Margaret in maturity. Never Molly. That name was a fiction invented long after her death - possibly because it had a better 'ring' to it for magazine journalists, scriptwriters and lyricists.
In Denver High Society, she was known as Mrs. Margaret Brown, never known as Molly, until Debbie Reynolds played her in "The Unsinkable Molly Brown", oddly, most Denver residents today refer to her as Molly!