In the circles I have "traveled" in over the years one can be Mormon but not necessarily LDS. A member of the Church of Jesus Christ of Latter Day Saints is LDS. Someone who is LDS lives the religion as it was intended. They love the Lord and do not consider themselves to be better or more righteous than someone else. They are not exclusionary but rather love and embrace those not of their faith rather than ostracizing them. They truly live the Gospel, they embrace the truth and serve those around them without judgment or qualification.
A Mormon ( or Mo or Mo-Mo) is a member but think they have the golden ticket to the Celestial Kingdom and have a holier than thou attitude. Or they can be a naive individual who follows without question, such as a sheep would. They love the Lord but love what people think about them more. They do all things according to the letter of the law but oft times neglect the spirit of the law.
To me, a Mormon is someone who accepts the Book of Mormon, Joseph Smith as a Prophet, and the Restoration. Not all believe the same things. Some are members of the LDS church, others aren't. Not all members of the church are Mormons.
Here is a quote from Joseph Smith, that for me, sums up the attitude:
I am referring to the fact that a lot of members of the LDS church are offended when the media talks about "fundamentalist Mormons." LDS members don't want to called Mormons, but don't want anyone else to be associated with the name either.
The point I have been trying to make is just wanting a title doesn't make that title yours.
Like I tried to point out, can I call myself Catholic, just because I fit a couple of the same beliefs as Catholics do, doesn't make me Catholic in any way. The reason this doesn't work is because of the definition of the word Catholic means I'm part of the Catholic Church. Now I can even say I have formed my own VERSION of the Catholic Church, one that doesn't believe in the Pope or something. But I'm not the one that can change the definition that society has given the term Catholic.
Because I don't follow that Definition anymore, then it is me that is different then others.
The same holds true for Mormons or should be. Mormons is another name for the Church of Jesus Christ of Latter Day Saints. As much as some other church tries to Claim that title, it doesn't work. They aren't the same church as much as they want to be. They can say it all they want, but in the end they aren't use the correct definition of the word. As much as they want to be part of it, the way society is, Mormons = LDS. That is the view for now. No matter how much people try to change it.
Now us being Mormons know and see the difference in these other churchs, and can see how people might show claim to the name (a belief in the book of mormon). But that doesn't change the definition of the word, or the view that society has of the term.