I am not fond of the term bullying by exclusion. I think it's too open to varying interpretations. It don't think we should be guit tripping children and labelling them as bullies for simply exercising the right to choose who their friends are. It's not how we socialise as adults so why do we expect it of children?
No-one tells us we MUST invite our annoying next door neighbour to our barbecue because she's lonely, even though we can't stand her. No-one tells us that if we are going to invite our two good friends from work to our wedding then we MUST invite the whole office including the creepy bloke that says weird things and always invades our body space, as much as he'd dearly love to be invited.
I appreciate that children need to learn to be tolerant and accepting of everyone in large group settings and not to be actively unkind to any other child, but doesn't mean they should be obliged to accept another child into their clique or obliged to play with them 1-on-1 if they don't really want to and perhaps have good reason to not want to.
It's sad for children who are not very socially confident, or are still learning how to behave in social settings so they may end up alienating other children or just being forgotten about. Of course I wouldn't tolerate children seeking another child out just to taunt them about how they are not wanted in the group. That would be bullying. But simply ignoring a child most of the time and not actively inviting them into games or a clique is not bullying in itself.
Children can sometimes find it hard to distiguish between the social cues that say 'I'm being pleasant but non-committal towards you. I don't want to hurt your feelings but I really don't want to encouarge you either.' and the ones that say 'Join our game, be in our gang.' So sometimes if they don't take the hint it might get to the stage where they are told bluntly 'this is our game and you can't play.'
If a child has previously been included in a group, whether organically or because adults have engineered it, and the group then decides they've had enough and want to oust that child from the clique, that can hurt a child's feelings a lot, I know. But sometimes there are valid reasons for it and it's a learning curve for that child that they might need to work on their social skills a bit. Be less bossy. Play less roughly. Be less needy. Don't talk over people all the time or tell them they are wrong/stupid, or lecture them constantly. Don't be a telltale or a whiner. Don't tell grandiose lies. Don't be a crybaby. Learn to be a better loser. Don't try to monopolise one friend and sulk if they want to play with someone else sometimes. Regulate your emotions better. Learn to share. Or sometimes simply do not be a bit of a bully yourself.
There will always be a reason, even if the only reason is that you are bit boring and they think they don't have anything in common with you. You haven't done anything wrong exactly, you just don't float their boat. However hard that might be to accept for the person not being included, it's perfectly okay for all of us to choose our friendship group (and limit access to it) on that basis. It's a good enough reason for me as an adult, so why is it not good enough for a child to do the same?
If that ever happened to my child, I'd be saying 'it's a shame, but these are not your people and you should focus on finding other children who are. You can't force someone to be your friend if they don't want to.' I don't think parents do their children any favours or teach them anything about resilience by wading in and accusing other kids of 'bullying' for not wanting to have their child around all the time.
In schools and hobby groups of course the adults in charge should try to make sure that no child is completely left out, but that doesn't mean that every child has the right to insert him/herself into any friendship group they like the look of whether they are wanted there or not. Imagine if we had to live like this as adults?