Incorrect Caller ID for Company Calling Me

Full Disclosure: I am a Telephony Consultant. I also happen to be a Cox Customer.  One of my clients had a name change to their organization some years ago, however, their old name is displaying as their caller i.d. - only for Cox customers though. Their new name does display correctly for Comcast and ATT customers.  What is the appropriate process for them to correct that?  They have numerous numbers (about 12 of them) for all of their practices. I don't know if they are all incorrect but anyone calling my office from their various sites is displaying incorrectly but when I was on ATT - I was getting the correct ID for them.  They have changed their Billing Name with ATT which I also had the understanding would trigger the Caller ID updates. Please point me in the right direction to get this accomplished.