My son does the right thing at home, because he wants to, not due to fear or expecting a reward.
At school, he thinks its a bit of fun to have a bumble bee on a chart they move forward for particular things, when they reach the end they get time on the ipad. He gets some time on it today.
The teachers didn't seem to think it was a problem or would confuse him. Home is different, and it should have a fairly large impact on how he reacts to rewards and punishments at school. Its not over the top on rewards.
He knows school is different, and they have their rules there, just as we have boundaries at home.
At school they consider wrestling not a safe game, and have to protect everyone. At home if he was to play with his brother like that, and his brother was happy to do so, I wouldn't intervene.
Another rule at school is no hat no play (unless under a covered area). We think vitamin D is important, so when with us, or before school he knows its OK not to wear his hat so he can get vitamin D. He has not asked why school is different, and he has been doing pretty well doing what the teacher asks.
I think for the wrestling and being warned 3 times for it, is because one of the other boys is the ring leader, and encourages it. So I can only remind my son about the school rules and the next stage in discipline if it continues. But the teacher is keeping an eye on the 3 of them. If there is one child who is continuing to encourage this behaviour, then I will pursue it, as I imagine the parents will be dealing with it differently to me.
I do believe home, and the rest of the world can be kept separate. School has rules that get warnings if broken, then punishment. Home we have boundaries, and if those aren't met the results aren't a punishment but natural consequences.
For example if he was busy doing something, and I went and asked him to get ready as we were leaving for school soon, and he continually kept playing and not getting ready to leave. I would remind him that he will not have enough time to play at the park, and that he may be disappointed if we run out of time. And yes if he didn't want to get ready but keep playing, then fine, he chose to miss the park. But that would be my last warning, as I will not let him get to school late, and I am driving, so after that I would say we are going.
Maybe not the best example, but its something I could think of right now. I don't have a real example right now.