"The name “Mormon” has become an unofficial but inoffensive nickname for members of The Church of Jesus Christ of Latter-day Saints."
We don't mind the label now as long as it is used properly.
The Associated Press Stylebook notes: “The term Mormon is not properly applied to the other ... churches that resulted from the split after [Joseph] Smith’s death.”
Journalists don't always follow their own rules and that is usually where the confusion comes from.
To be a member of The Church of Jesus Christ of Latter-day Saints is to have an actual membership record. It is NOT a vague feeling of identification or affiliation. You either have a membership record or you don't. You either are a member or you are not. There is no gray area.
Real Mormons are those who have an actual membership record with the Church. If your membership is cancelled either by yourself or the Church then you cease to be a Mormon.