I’m officially a member of the LDS Church (the Church of Jesus Christ of Latter-day Saints), though my fidelity actually lies elsewhere.
“LDS Church” is a standard, inoffensive name for the church itself, and “Mormon” serves as a decent adjective for the people, culture, and history of all that relating to the LDS Church. The term “Mormon” began as a pejorative by our detractors during the founding of the church, but like so many titles the Mormons began to embrace the term. It’s why we speak of “Mormon food”, and “Mormon folklore”, or the “Mormon Trail”. There are some sticklers who insist that it’s inappropriate to use the term “Mormon” for anything besides the Book of Mormon itself (or the ancient American prophet therein for whom the book is named), but they certainly are in the minority.
What is insisted upon pretty universally is that one does not refer to the “Mormon Church”, for the same reason the Lutheran poster above mentioned: Mormons believe they are following Christ, not a person named “Mormon”. In reference to the church itself, LDS members typically insist on the acronym LDS or the full name of the church. There are some in the church who are trying to push the use of “Church of Jesus Christ” as the name used in secular publication and discussion, which I find entirely pretentious. It’s one thing if we were to use this shorthand among ourselves in our meeting, but in mixed company it will at best foster confusion and at worst disdain.