Anyway, I just checked my tape of the original video release (which was 1986, not '85), and it doesn't have any of those Paramount Television logos. Since it was an original video release that had never aired on television at that point, it bore the logo of Paramount Video instead. You can find the logo they used on the Paramount Home Entertainment page of the logo site:
It's under "3rd Logo (1982-88)," the "Growing Mountain" (it's the video clip on the left-hand side), with electronic music by Richard J. Krizman. It appears at the beginning of the video after the copyright warning; then we go to Roddenberry's 1986 introduction; then we get the episode (cut together from the B&W "Cage" and color "Menagerie"). The end credits of the episode conclude with a Desilu copyright notice but no separate logo card or music sting. Then we get Roddenberry's closing statements, and then the "Growing Mountain" logo again.