OP you sound like a good mum with a fairly typical young boy.
I agree it’s is helpful to label the behaviour not the child, but it’s important to recognise that certain behaviours are unhelpful (eg laughing, commenting and disrupting when the teacher is handling another incident) and certain behaviours are unacceptable (eg having a meltdown when given a minor sanction).
OP I love martial arts for kids who need to learn self discipline.
Also sit your kid down and go through a list of likely behaviours and consequences both good and bad. Make him work out what the consequence might be for each bad or good behaviour on the list.
Make it really clear that time outside the classroom is “breathing space” as Teacher can only deal with one thing at a time. Then if he is kicking off outside the classroom he has turned “breathing space” into an incident which is unnecessary because the whole point is that the breathing space is his moment to learn to calm down and say “sorry Miss”. Just a quick thing, no drama. Once he has learned to do that, teacher will invite him back in the classroom.
Give him a list of things he can silently do in the breathing space outside the classroom to calm himself down so he can recenter. Practice this with him. For example - close his eyes and put his arms out and stand on one leg for as long as he can. Stretch his arms up to the ceiling and draw the biggest circle he can with his arms. Stretch his fingers and then form fists 20 times. Take a deep breath and hold it for a count of four then blow it out like he is blowing a candle as slowly as he can. He should repeat these as slow as he can, seeing if he can drag them out so he only has time to do a couple of cycles before teacher comes out to him, see if he can beat his own record each time. The slower he completes each the cycle the better he is doing.
As soon as the teacher comes out of the class to fetch him he has to instantly stop his routine, make eye contact with the teacher, say “sorry miss for being disruptive”. That’s all. He just learns this rote behaviour and practices. You can even practice on silly things at home - make a game of it and you join in. Do something very noisy and active in the living room eg play keeping a balloon in the air. If someone lets it fall on the floor, they have to go in the hall and do the calming routine until the other person comes to get them. Yyou’re trying to simulate being very excitable and then quickly learn to tamp it down using self control techniques.
Anither good game to play is musical statues - going from hyper to totally still is very tricky at this age. But it helps them learn to switch gear really fast when external conditions are imposed on them.