Actually it was a GOP coup. Per wiki:
“In the summer of 1997 several House Republicans, who saw Gingrich’s public image as a liability, attempted to replace him as Speaker. The attempted “coup” began July 9 with a meeting of Republican conference chairman John Boehner of Ohio and Republican leadership chairman Bill Paxon of New York. According to their plan, House Majority Leader Dick Armey, House Majority Whip Tom DeLay, Boehner and Paxon were to present Gingrich with an ultimatum: resign, or be voted out.”
Technically you are correct. But the GOP is not the maker of "public images". That's the job of the Dems and the MSM. The Pubbies were just reacting to the good job done by their political enemies in defining the supposedly evil and reckless Newt.