I wonder how much regional culture matters. I grew up and spent several decades of my adult life in New York State and the surrounding area, but since I got married I have lived on the west coast. Back East I was considered kind of quiet, and here I am considered loud. I mean, just in general- just when talking to people.
As far as cheering for the other boys they are in rotation with, my boys said they do not usually do this, because (according to them) some kids do not like being cheered for and they do not know what that particular boy might want. When it is someone they know, and know he likes to be cheered, teammate or not, then they cheer, otherwise they do not. They will of course always say good job to the kid after a kid goes or whatever. They are supportive. They just do not cheer. For them, not cheering for the other kids is about being worried they might rattle the other kid, not because they do not want to cheer anyone who is not on their team.
Now that I think about it, I think they also take a cue from their coaches, who tend to be very quiet at meets. Weirdly the better things are going, the quieter their coaches get! They very rarely show emotion on the competition floor. This would not be my way, but I get it, this is their style, and my kids expect the quiet words and not the big reactions from their coaches. But now that I think about it, this may be why my boys do not feel as comfortable with cheering loudly. Again I think too much quietness at meets heightens tension, and loud cheering helps everyone relax. And I have told my kids this, and they pay as much attention to this advice as they do anything else I ever suggest about gymnastics.
But, there are some boys on the team who clearly love to cheer and cheer loud and let the chips fall where they may, and that is more my style.