When a number is "unpublished" that merely means that your telephone company (Verizon) does not 'publish' it in their printed telephone directory.
Displaying your name and telephone number on someone's Caller ID display has nothing to do with whether your number is unpublished or not.
You can ask your telephone company to permanently "Block" the displaying of your name and number for every call you make; or you can block any selected call by dialing *67 before you dial the number you want to connect to.
Remember that anyone with Caller ID also has an optional feature called "Anonymous Caller Rejection". This feature will specifically prevent "Blocked" calls from being connected.
In such a case --if you have had your telephone company to permanently "Block" the displaying of your name and number for every call-- you can temporarily override the 'Block' by dialing *67 before you dial the number you want to connect to.
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