For example let's say someone said "Dad just got home" as joke but the father hadn't actually arrived, or "He told you to take the trash out" but that was a lie and didn't actually happen. How would such statements be classified?
Objectivity means being impartial, trying to correct for any cognitive bias, or for mood, say. To make an objective statement is to take a neutral 'everyman' perspective, not related to personal and subjective experiences and views. Objective vs subjective distinctions belong to discussions of bias.
The statements you mention belong in a different context: finny vs not funny; true vs lie. So that's how you would classify them. If say a court case happened around the events, police and lawyers would interview everyone involved and look for evidence, to build an objective account of events that was independent of any one person's account.