The term Mormon was first used to describe LDS Church members by their critics. It was a derogatory term and eventually they just decided to make it their own. So at some point in history while this transition was underway, it must have been awkward for non-Mormons in Utah to use that word.
Maybe some guy’s at a restaurant talking about how those damn Mormons woke him up to proselytize at him. A Mormon couple overhears him and get pissed: “HEY! That’s OUR word. Only Mormons can call another Mormon a Mormon.”