You are correct, it was founded 24 Dec 1865, by six Confederate veterans who met in the office of Judge Thomas M. Jones, viz.: James R. Crowe, Calvin E. Jones, John B. Kennedy, John C. Lester, Frank O. McCord, and Richard B. Reed.
Forrest never served, nor was he the leader at any time - he was elected in absentia and never served. The only 'official' proclomation he ever made was for it to disband, and even then, that was written by Capt. John W. Morton.
What it became is just another unfortunate stain on American's shared history, and it was well established as a white supremacist terrorist group by the time Byrd came around.