If you notice that all of your notification emails and text messages from Acuity have one of your client's names attached to it, what this most likely means is that you have saved scheduling@acuityscheduling.com in your contacts as one of your clients.
To fix this, search in your email program's contact list. Make sure you do not have a name other than "Acuity Scheduling" associated with scheduling@acuityscheduling.com in your contact list.