get the best of them, I don't think anyone particularly cares or is offended by it. Each group has their preferred formal name.
The former RLDS long ago decided to deemphasize the Book of Mormon and try harder to fit in with mainstream protestantism. Nobody really cared because they are very insignificant in number. If you call an ordinary CofC member a Mormon, I doubt that they would be offended. They may tell you their preferred label (and I don't even know what that is)...and then get offended if you make a point of ignoring that, but otherwise...not so much.
For most people inside and outside, "Mormon" primarily refers to the Utah-based church currently led by Nelson--due simply to the fact that it is overwhelmingly the largest and wealthiest of all religious groups that trace their origin back to Joseph Smith and his Book of Mormon.
I don't believe they're offended by it, but they don't use it to describe themselves.
They're way more laid back than TSCC is. Changing its name from RLDS in 1999 was to distance itself from the Mormon church. They still use the BoM, but treat it as an "inspired" work not in the same vein TSCC treats it. It also uses Smith's translation of the Holy Bible, which I found laborious, and more proof of his dishonesty.
They also don't revere him as a prophet the same way TSCC does. They consider he was "inspired," but don't consider him as equal to Moses or at the right hand of Jesus in the eternities the same way Mormons have put him on a pedestal.