I haven't gor children of that age of step-children so I may be way off the mark but it might be worth having a talk with him that appeals to his intelligence.
That he's smart enough to understand that different houses have different rules and his mum has the rules that suit her and you have the rules that suit you (nothing wrong with his mum's rules). In your house he needs to follow your rules and remember them for next time he visits.
I may be wrong but I don't think that 10 is too old for some kind of chart. points on for good and off for bad and if he has enough points he gets X and if he loses too many points he doesn't get Y or similar.
Positive reinforcement is always great (it works on my DH so I think it works on boys of all ages lol). Try and 'catch him' doing the right thing and praise him. Even he's only done it for a few seconds get in there and praise him before he stops.
My niece decided to stop brushing her hair a year ago. My sister was arguing about it with her every day. I spoted it brushed once and told her how pretty she looked with her hair like that and the next time I saw her she had brushed her hair again to get the compliment. maybe you could rope a friend in to do something similar - sometimes praise from someone else can be even more effective than parental praise.