In "Havisham," the one thing that defines our speaker's life is her wedding, or, to be more exact, her lack thereof. Miss H was jilted by her fiancé and chooses to spend the rest of her life surrounded by the symbols of the wedding that never was – her wedding dress, the wedding cake, etc. She pines for a marriage that never happened and creates a world for herself in which time stands still. Marriage is the one thing that Miss Havisham wants and the one thing that's impossible for her to get. Especially since she hasn't showered in decades. This isn't how you snag a new man, ladies.
By referring to herself as a "spinster," Miss Havisham emphasizes how pathetic her life is. She thinks of herself as a failure.
By referring to herself as a "spinster," Miss Havisham takes power over her situation by facing the facts.