Has he been to a pediatrician about this issue. There can be physical reasons for "accidents." I think this should be the first step. One cause could be an immature digestive system. Another could be his diet. Children can have IBS. Or their system is just more sensitive to some foods than others.
I have IBS, irritable bowel syndrome, and I can tell you that there have been times, as an adult, that I can't make it to the bathroom in time. I could understand that he might not want to get up and run, thus drawing attention to himself.
My 8 yo granddaughter has had a few accidents for the same reason. The bathroom is too far away. She is instantly mortified and we do go to the bathroom and together clean up. I treat it as just one of those things. No big deal.
If there are no physical causes then I'd explore the possibility of emotional issues. Keep a diary of when this happens and what is going on at the time including what he's eaten. You may discover that he doesn't want to interrupt his play. If that's the case I would talk with him once about it in a positive way. If this is the case I'd have him clean up his own mess.
I'm surprised he's not embarrassed at school. Does he have developmental and/or social issues.
Is it possible he's not embarrassed because you are?
Could this be a will power situation or control issue with him. You suggested that it was possible he was toilet trained too early. IF so that could bring up those issues. He can be doing "pay back" without even knowing that is what he is doing.
I'd also want to know how the school handles this situation. Is there response a positive one that also gives him responsibility for managing his bowel movements? Having accidents results in attention. When we aren't getting the attention we need we often revert to negative behavior. The saying "negative attention is better than no attention at all" is true. Or he could be getting positive, helping attention both at school and at home, as he should be, but he may have come to rely on poop as a way of getting attention.
If attention seeking is the reason then be matter of fact about the accidents, having him clean up for himself. Not in cold water. That is punishment, not discipline. The goal is to help him learn how to manage his bowel movements in a way that he controls. The boy in cold water may manage them but it's from fear and being uncomfortable. This hides the "problem" which may come out in other misbehaving ways.
It's the next step that helps, if he's attention seeking, show him positive attention everyday. Stickers and treats may be good but they may not actually result in good parent child time together. And it only addresses the issue of poop. He needs to be praised several times during the day for the good things he does. Sometimes one has to really look to find these things. Hopefully your son isn't acting out in other ways.
It's also important to spend time just with him doing something he enjoys. Time when poop is not a part of the conversation. Focus on good behavior. When negative behavior shows up, redirect to positive behavior without trying to teach anything by talking about it.
When I adopted my daughter at age 7, she had many negative and highly emotional behaviours. Looking back I realize I spent more time working on the negative behaviors then praising her good behaviors. I talked alot about what she should and shouldn't do. As an adult she tells me she wishes I'd praised her more by telling her she'd done good and saying I'm proud of you, I'm glad you're in my family etc. General statements of praise are so important. They are working with my grandchildren.
I think part of my daughter's difficulty with self-esteem is the result of my trying to correct her behavior. Of course her self-esteem was already as low as it could probably get. And I did do many helpful things so that she is a successful adult. But I wish I'd been more proactive and praising of even the small successes. The two of us spent too much time being angry with ourselves and with each other.
I suggest that you are embarrassed because somewhere in you mind (perhaps not consciously) you feel that you are responsible for the way your child acts and what he does. For example: "If I could only find a way to stop this pooping I'll have shown that I'm a good parent and my son will be seen as a good kid."
I know that I felt this way with my daughter. I wanted her to wear nice clothes that were always clean because I thought that the way she dressed reflected on me. My grandkids can be dirty and wearing raggedy clothes and I don't pay any attention. I'm not responsible for the way they look. :):):):) It's their mother's responsibility. Unfortunately she now has the attitude that I had and wants her kids dressed in nice clean clothes and will have them change before leaving the apartment or will say to me, "Mom how could you let her leave the house looking like that!"
I reread your post. I'd stop with the reminders, etc. at home. Allow him to take responsibility. Pay as little attention to the poop and accidents as possible. If he needs to go out in public tell him that you don't want to be with the stink. Therefore he needs to clean up before he can go. That sort of thing. Focus more on how you feel about the stink than on trying to get him to stop with the poop. "Son, this room is too smelly for me. Please clean up or go to your room so I don't have to smell it."
This puts the focus on the poop and not on him. Be careful to not ever say anything critical of him as a person. Be sympathetic to his problem but don't blame it on him. You could ask him if he wants your help with solving this problem but if he says no or is unco-operative then back away. Very difficult to do, I know.