As luck has it, I have around dozen episodes in my 16mm collection.
Unfortunately, the Paramount and/or Desilu logo is cut from all of my 2nd season episodes if I recall correctly. However, I will pay attention to these the next time I get a chance to screen my films. I MAY have it, because I'm almost 100% certain I've seen Nicholas's version of the 1968 logo with the Desilu music before. I have not found it on any of the official Paramount tapes yet. It could be on the first 5 Volume set released in 1980 called "Television Classics" -- as these were earlier film transfers than the 83-84 transfer.
I also have a number of 16mm "film chain recordings" on video tape from the very early 80's right before the switch to distribution on video. Often times, the end credits were cut early on these as I recall.
As a matter of historical fact: The sale of Desilu to Paramount was official on July 28, 1967 -- all 1967 episodes, starting with "Mirror, Mirror" should have a Copyright (c) MCMLXVII by Paramount Pictures and Norway Corporation underneath. Norway was, of course, Gene Roddenberry's production company and is always present on the correct logos AFTER 1966. The 1966 Desilu logo had no copyright originally-- which actually caused all episodes prior to "Mirror, Mirror" to go into public domain until they were reclaimed by Paramount in 1978 when the Copyright Act of 1976 went into effect. Hince, the 1978 copyright was added to earlier episodes.
All 3rd season episodes for certain should have the zoom in style Paramount Television logo with the orange/yellow background -- this logo should not be on any other season other than the 3rd -- the music was recorded by Wilbur Hatch after the 2nd season had aired. Therefore, it couldn't have been on any 2nd season episode.
All 1968 episodes of the 2nd season should have the Paramount logo with the Desilu music -- that should be (latest to earliest): Assignment: Earth, Bread and Circuses, The Ultimate Computer, The Omega Glory, By Any Other Name, Patterns of Force, Return to Tomorrow, A Private Little War, The Immunity Syndrome, A Piece of the Action, and The Gamesters of Triskelion.
All 1967 episodes AFTER the sale of Desilu should have the Desilu logo with the Paramount and Norway byline as mentioned above: The Trouble with Tribbles, Wolf in the Fold, Obsession, The Deadly Years, Friday's Child, Journey to Babel, Metamorphosis, I, Mudd, Catspaw, The Doomsday Machine, The Apple, and Mirror Mirror.
The 1966 Desilu logo with no copyright should be on: The Changeling, Who Mourns for Adonais?, and The Amok Time.