The first reason I will continue to use the term Mormon is because Joseph Smith himself accepted the nickname Mormon and played around with the term and analyzed it through the study of languages: and said that when you do that you can say that Mormon simply means "more good."
This is the main reason why I will continue to use the word Mormon, because it represents what being an Emergent Mormon is all about, that is appreciating that Joseph Smith's theo-philosophy is more good.
Second, the term "Mormonism" is canonized in scripture in D&C 135: 7. Enough said.
Thirdly, the term Mormon distinguishes one's cultural heritage and personal philosophy as LDS from Catholicism and Protestantism. In a Midnight Mormons episode (for November 2021) they discussed changing their podcast name and removing the word Mormon. A commenter named Liberty LLama responded, "I think that calling yourself a Mormon really draws a line in the sand of where you stand, and what you believe. When we call ourselves Christians. The assumption is that we believe in an immaterial god, and a three in one god." That is exactly right.
When someone says to me, "I'm a Christian." I really don't know what they mean. Are they an Evangelical Fundamentalist Christian? Are they Catholic? Are they an Episcopalian? So nowadays that term really does not clarify anything specific. The one thing that phrase "I'm a Christian" does mean to me is that they likely hold the view that God is this vaporous no-thing, there is no feminine Divine, no Heavenly Mother or female angels (only a single dad and celibate son and all male angels), and people are consigned to eternal torture for thoughtcrime. As that is what is usually entailed with the designation of Christian as I've encountered it. So saying "Mormon" avoids that with Joseph Smith rejecting an eternal hell with his Three Degrees of Glory and his other universalistic theologies and rituals. It makes it clear where you stand philosophically and theologically. And yes it's true that Mormons do have a different Jesus than the sectarian creeds. There's even an academic book called The Mormon Jesus: A Biography by John G. Turner.
As I cover in this blog, Joseph Smith's "more good" theology moved away from the despising of the body found in puritanical sectarianism. To use the term Mormon not only calls to mind a character in The Book of Mormon (scripture other Christian sects don't have), and distinct LDS theologies different from the sectarian Creeds; but also calls to mind a Life Philosophy that embraces joy in the body and a sex positive attitude: that grew out of Joseph Smith's jovial nature and "people person," joyful, personality; which comes through in his thinking and philosophizing; which is distinctly different from the personalities and thinking processes of Augustine, Luther, and Calvin.