It sort of sounds like SS is never sure what will or won't get him in trouble.  He was so concentrated on fear of getting in trouble for not taking his snack and instead got in trouble for lying about whether or not he took his snack.  He probably feels like he is darned if he does and darned if he doesn't.  Are the house rules consistent and does he understand them? 

Does DH actually expect him to follow the rules when you're not around?

This is why I've stopped enforcing the majority of rules in my house when they don't impact me/my household. I noticed SD was acting like a timid, scared dog around me. For example, I would come downstairs and there'd be a plate of half-finished food on the table and she'd be sitting on the floor playing than she would run over all afraid and clean up her mess. I hated feeling like this big scary monster in my own house. I realized eventually that DH doesn't remind her to wash her hands/clean her plate after she eats. 

He doesn't do it because he doesn't care if she follows the rules, he has ADD and doesn't notice. But I was the ONLY person in her life being consistent with rules... And they're simple rules. Wash your hands/clear your plate when you're done eating. Pick up your toys at the end of the day. Take your clothes out of the bathroom after you shower. Don't try to grab chocolate/treats if you threw out your lunch/dinner. Little things that should be expected. But yet I was turned into this scary monster in her mind because I was the only person who cared about them. 

This really started affecting me, like seriously affecting me, because she made me look abusive. When we were around family or friends, she would ask me all timidly if she could eat a treat someone handed her or ask me all timidly if she could play with her brother. Little, strange things that made me look terrible. Like yeah, of course, it's not like I would ever disallow those things? I just expect you to wash your hands after eating/before playing? And not eat treats after throwing out an entire meal? 

I will add, it's funny to me because I know DH's family gets the wrong idea of me because of SD's little timid act with me. But they all refer to her as the "best behaved kid" in the family - they praise how great her manners are, the fact that she doesn't make messes/bug their pets, etc. and how great of a guest she is at their homes. Yet they fail to realize it's the big bad stepmom who is to thank for that. It's my small, simple, common-sense rules that she's terrified of that make her such a great kid. But whatever. I've just really stepped back and stopped caring. As a result, her room is a pigstye, her clothes/toys are everywhere, and it's not my problem. 

Ash, just wondering if the punishments are always the same? Like, take away tablet for everything? The punishment should fit the crime, imo. As in, if you lie to me, I won’t believe you for a while. I’m not going to take away the tv or the tablet... maybe try making it a point to question his next ten or so actions until it drives the point home. It’s adult-training, because that’s what you’d do if an adult lied to you, but you just add in a little theatrics so that the kid understands the actual consequences of lying.

Youre doing a good job, btw... it sounds like you care about him a lot, and he’s lucky to have you help guide him.  

What's up with the snacks? A natural consequences of not taking a snack is that he will be hungry.  So stop reminding him (but first find out why he won't take them, does he not like them?) and let him be hungry if he forgets.

Yeah not sure about punishment for not taking a snack?  Why?  If he doesn't have a snack then he might be hungry.  No harm, no foul.  Eventually the light bulb will turn on and he'll start taking snacks.  I understand that he lied which, in itself, is wrong.  However, you are pressuring him to eat food, which can cause issues in having a healthy relationship with food.

Pick your battles.  If he's eating breakfast, lunch and supper than there isn't an actual problem.  Save grounding for big stuff like not doing homework or having a rotten attitude or talking back.  Not eating the apple, banana or granola bar is just not a crime.  I don't get it.