Is a wife who set herself on fire to scare her husband, and then died a month later from the burns, considered to have committed suicide Islamically, or is she merely sinful for harming herself, her husband, and her daughter?

There is no difference between the two questions, as the doer did not intend suicide, but rather intimidation. A suicide is one who intended to kill himself, whereas one who engaged in a fatal act without intending suicide is not considered a suicide, but is sinful. Accordingly, the wife is not a suicide but has committed a great sin, and it is advised to avoid quarrels.