Puzzling. The Family Visitor attracts appeal rights which can be exercised in-country if the applicant is a non-visa national. You arrived as a Family Visitor, but then opted not to disclose it and attempt landing as a visitor instead. Because of that you forfeited your right to appeal and got bounced.
If I got it right, your intentions were lawful, but your strategy was rooted in not knowing the difference between a visitor and a Family Visitor. This would be because for non-visa nationals, the Family Visitor is not relied upon as much as it is for visa nationals and so there is a lot less information about it. I am surprised they didn't allow at least a TA despite your having initially mislead them. They changed the TA rules in November, so there might be confusion about that at the ports. I hope you checked the box on your VAF1.
If they bring it up in your interview, you would probably stress that your intentions were lawful and you met all of the other criteria; and since the information about Family Visitors for non-visa nationals is so scarce, you were unsure about how to frame it and made a judgement error. And since then you have taken steps to learn about Family Visitors - it would be important to add that point. My sig line has a link for US visitors.
If they don't bring it up, then I wouldn't go introducing the topic.
