How to remove a real-id from a friend in BattleNet

Click your account character name (located below the search bar in the forums), it should have a down arrow next to it. Click "Manage Characters" and select your Diablo 3 game name and posting again.

That answer is pretty much it yes - the major difference between Battle Tag and Real ID is that giving out your Battle Tag is not exposing the email address that you have associated with your Battle.net account.

When Real ID was first implemented a lot of people complained that in order for people to add them to their friends list, they had to give out their Battle.net username (email address), which was a potential security risk.

Blizzard's answer to this issue was the implementation of Battle Tag, which is a name that you choose with a randomly generated number on it (ie: YourName#1993) which you can then give out to allow people to add you to their Battle.net friend's list without exposing your email address to them.

In addition to this, as pointed out in the article you have linked, people added to your friends list with their Battle Tag will only show their Battle Tag, instead of showing the name they have associated with their Battle.net account.

The reason people ask for your Real ID still is probably out of habit, or due to them being unaware that Battle Tag is the same thing but with better privacy - remember that giving out your Real ID means you're giving out the username to your Battle.net account.