Just to clarify this…
FreePBX configures Asterisk to send both Caller ID Number and the Caller ID Name (“CNAM”) that you specify. However, in the U.S. most carriers dump the CNAM that you send, and then use the Caller ID Number to lookup the correct name from a database. Looking up names in the database usually costs money (a fraction of a cent), so some VOIP carriers won’t use the database or offer you the option to use it or not. All major carriers (AT&T/Verizon/etc) and (I think) all wireless providers consult the database for inbound CNAM.
Some carriers can update this database for you if you want to change what appears. Some carriers charge a fee to update the database with new information (VOIP.ms charges $10.00 and you have to open a trouble ticket) and some do it for free (Flowroute let’s you do it via the web portal and AT&T does it automatically when you sign up).