Please note that this issue back in the day when land lanes were super popular. I use the caller ID to find out who is calling before I pick up the phone. Nevertheless, your concern is valid and Fido can certainly make the changes for you.
If I reemmber correctly we used to be able to change this setting on your cell phone so that other party will only see your name, your phone number, or no name no phone (private caller). I don't know if this feature is available in late model iphones; however, there is a setting in my android phone. It is under "Call Settings" --> "Advanced / More setting" --> "Additional Settings" --> Caller ID --> (Network default, Hide number, or show number).
The reason I suggesed hiding just the phone number and not the name so that people will know how called instead of just displaying "PRIVATE CALLER" or something similar. Usually I don't answer phone calls of that nature. If you are trying to get a hold of me (for example), you might get the voicemail instead of me picking up the phone.
Hope it helps and good luck.
