If you go to the Family Manager dashboard, are you able to click the name of the user with the incorrect name and edit the name to be correct?
The URL provided works great for updating the family name, or updating an individual's name including the family manager. But that's the only place it helps. It doesn't actually change the account name.
I came on this thread to see how to change the account name.
My personal/family account is also linked to a Work account; alphabetically my personal account (based on email address) appears at the top of my folders list in my work account. I get that my LoginID is my email. But it appears that my LoginID is also my Account Name? I want to share my personal account with my work account using a specific account name such as zSandy - so the personal account shows at the bottom. It would also help when associating an account with an existing folder; currently my personal account and folders appear at the top of the drop down list.
There isn't a way to change the Account Name for personal accounts (usually labels will just use your LastPass username, which is your email), could you try editing the folder name for any shared folders (such as in your linked accounts) to see if you can change the folder name to what you'd prefer it to be?