Don't worry... some kids hit, pull hair and push or slap.
He is 2 years old...they DO this.
At this age, they do not have FULL "impulse control." Full impulse control is not developed until about 3+ years old.
So if you expect him to just "stop" it forever... you will be disappointed in him all the time.
And no, Time-Outs will NOT always work... they get desensitized to this too... because it is used against them ALL the time... it becomes useless.
Sure, he seems to understand when you "explain" to him... but this is only the beginning of their socialization and learning social "rules." A child will spend ALL their childhood learning social protocol. Even teens have to learn this, and they are not "experts" yet either.
Okay so, the YMCA should know full well about child development PER each age set. Now, if your son is "biting" just willy-nilly...then maybe like the other responder said, maybe he is teething. BUT... you want to make sure that other kids aren't bugging him either or provoking something in him to do this.
All kids have "triggers." So try and see if there is something that triggers your son to bite??? My son used to bite too... when he was teething. He was a nibbler. But it stopped. Some kids will do this when they are tired, or frustrated....
The point is... good you are showing him "how to" apologize to the one he bit. But a child this age is NOT going to be proficient at it, yet. A child this age is NOT going to just follow social protocol perfectly, at will. But, over the course of "teaching" him... also teach him about "empathy" how to care for things, how to understand what "hurt" means... and what not. He will not get it now, but OVER TIME.... this will be building a foundation in him... in how to "feel" for others and how to care for things. But for now, it is still not going to be understood perfectly.
If your child gets kicked out for biting...then it is because it is a problem for the other kids, for the group leaders, for the other Parents. No matter what, a child care organization has to do what they must, in light of their own rules too.
But don't ever directly usurp your son's self-esteem, in teaching him what is not acceptable. They need to learn boundaries.. but you know what, it takes time.
It's a phase. Just watch to make sure it does not escalate. AND, some kids are just "copying" other kids... and since your son goes to the YMCA...maybe he is just copying what other kids do. I'm sure your son is not the only one who has done this... so don't feel self-conscious.
It's good the YMCA IS trying to problem solve this, and the "buddy" idea sounds great. Try it. Just keep open communication with them...
Good luck,
Susan