So he has only been here a little over three months.
I think here lies part of the problem, not enough bonding time.
I had several stepdads growing up, hated almost every one of them, at least at first. They were outsiders to me, thinking they could tell me what to do. So your stepson prolly has that "you're not my dad" mentality.
You want respect from him, but you havent
earned it yet. Respect comes with time.
He prolly also has some resentment, and depression, from taking him out of a situation he was happy with. You have to get on his level too, with some understanding.
He shouldnt disrespect his mom, at all.
Yes, teenagers can be very frustrating, I know. Its a phase they go through. Mine is 22 now. They have to "learn things on their own", before they come around again.
If you were commanding him, in an angry voice, to come eat, I can see why he got hostile. Disrespectful.... yes. But again, you havent EARNED that respect from him. Im not condoning what he said though, that was pretty fucked up. I never had to deal with that with my son, and I never had a stepson to deal with. But I did have a stepdad that thought he was king shit, til I blocked his spanking, blocked his shove, and shoved him back, with a look on my face like "BRING IT MOTHERFUCKER!". He never touched me again. Glad my aunt put me through karate class lol. The guy never earned my respect, because he was a drunken asshole, always wanting to argue, and be in control. It wasnt til later, that he kicked the alcohol after my mom kicked him out, that we wound up getting closer. Too lil too late for him and my mom though, as she met someone else.
My mom kicked me out at 18, for calling her a bitch. Long story lol. But could legally do so. I hope it doesnt come to that with y'all.
I will also say, those Xbox friends, prolly contribute to his attitude. Some spoiled rotten kids with attitude out there. And they prolly put shit in his head. With corona keeping kids out of school, its prolly his only escape, and his only friends though.
If this was MY situation, I would first have a talk with your wife, about what you plan to do. Then I would take that son, somewhere he enjoys, or on a hike or something, where y'all can get away and have a talk. Apologize to him about how you snapped, but let him know why you acted that way. That it bothered you that he was disrespecting his mother, and that you were raised with manners and respect. Dont sound stern, sound understanding. See if he apologizes too. Let him know you love his mother very much, and that you only want the best for all of them. See if y'all can bond on a mutual understanding, and go from there.
Remember, for him to respect you, you need to earn that respect, and vice versa. Dont let him be a pushover though. I think after how you pushed back, he knows you wont fuck around lol, so he has a taste. But that either pushed him away more with a sour taste in his mouth, or he may of grown a lil respect for you, for not being a pussy.
I know this post was long, so I'll end it here, but could prolly say more lol.