The only option that you can control is that it will only show THAT character’s individual achievements earned. There isn’t anything that can hide them completely. And as to the website/armory - it’s temperamental on the best of days. Sometimes it takes some time to update to reflect the change.
Even then though, they could be referencing your gear/mog or any number of things. This is a MMO. It’s meant to be a social game. Achievements, pets, mounts and such aren’t considered “personal information.” It’s in a bit of a different situation from yours, but here is this:
