So the terminology here gets used differently by different people, but the view that moral statements can be true or false is usually called “cognitivism”, not “realism” (though there definitely are people who use “realism” for that view). My own personal preference is to define realism as cognitivism plus the metaphysical claim that moral properties are mind-independent (i.e. not grounded in facts about anyone’s moral beliefs or attitudes).
So the terminology here gets used differently by different people, but the view that moral statements can be true or false is usually called “cognitivism”, not “realism” (though there definitely are people who use “realism” for that view). My own personal preference is to define realism as cognitivism plus the metaphysical claim that moral properties are mind-independent (i.e. not grounded in facts about anyone’s moral beliefs or attitudes).