I sent out a group message to my family (from my new Verizon number) and they could not tell that it was me. I applaud their caution but did I overlook some setting? One that tells my friends and family who is sending.
Exactly. SMS and MMS messaging are phone-number based; to identify the sender by name, the recipient must have that phone number associated with a name in his contacts list.
Simply put, if the recipient does not have an entry in his/her contacts app that associates your name with your phone number, then their phone will not "recognize" you by name, neither for incoming phone calls nor for texts.