Ok, I have fully tested this out.
Here are the following points:
IF you are in a clan, and you have NOT shared your RealID with anyone else in the clan from your friends list, YOUR NAME DOES NOT SHOW UP ON THEIR SIDE, even though you see your name on your screen. I have tested this with my clan members and verified it.
However.
If you have shared the RealID with your friend, and both of you are in the clan, then your name may be visible to that friend.
The battlenet settings for privacy will not affect in-game settings either. In fact they have a note about this next to the privacy setting:
Settings → Friends and Chat → Hide friend’s Real ID (this only works in Battlenet and only for your friends, not you)
You can also access your profile privacy settings here:
https://account.battle.net/privacy
And view the RealID information page here: