The Principle Voices Coalition, based in Salt Lake City, said that members "strenuously object to any efforts to deprive us and others of the freedom to name and describe ourselves by terms of our own choosing."
Two weeks ago, the Church of Jesus Christ of Latter-day Saints launched a media campaign aimed at distinguishing itself from the breakaway sects.
The church said its effort was primarily aimed at clarifying the difference between the LDS Church and the FLDS sect, necessary because of widespread coverage of Texas authorities' raid on the sect's west Texas ranch.
The LDS Church said a poll showed more than a third of those surveyed thought the sect was part of the Mormon Church based in Salt Lake City.
"It's obvious we need to do more to help people understand the enormous differences that exist between our Church, which is a global faith, and these small polygamous groups," said Quentin L. Cook, an LDS Church elder, in a statement issued June 26.
He also said that "Mormons have nothing whatsoever to do with this polygamous sect in Texas," reiterating comments made several years ago by former LDS Church president Gordon B. Hinckley that there is no such thing as fundamentalist Mormons.
But the coalition objects to any attempts to limit use of the term.
"Fundamentalist Mormons have been referred to by that name since the 1930s, often by the Church itself," the coalition said in its statement. "We are proud of our Mormon heritage. Plural marriage is only one of the tenets of our religion, the Gospel of Jesus Christ as restored through Joseph Smith."
The LDS Church has asked that media refer to those who follow the original beliefs set down by Joseph Smith as "polygamous sects," the group said, but "most of us are not (and do not refer to ourselves as) polygamists."
The coalition said the statement had been authorized by the Apostolic United Brethren, the Davis County Cooperative Society, The Work of Jesus Christ and numerous independent fundamentalist Mormons.
The statement was not signed by the Fundamentalist Church of Jesus Christ of Latter Day Saints, The True and Living Church of Jesus Christ of Saints of the Last Days or several smaller groups.
The coalition said the LDS Church has experienced a similar problem with refusals of some Christian denominations to recognize Latter-day Saints as Christians.
"In many ways, we consider ourselves to be adherents to Mormonism (and Christianity) no less than were Joseph Smith, Brigham Young and John Taylor," the coalition's statement said. "What distinguishes us from the modern, mainstream Church is that we have endeavored to observe the original, fundamental precepts of the restored Gospel, while the Church itself has, since the early 1900s, repudiated several of them.