A quote from Courage but not from the TOS theme. The very short cues we hear during Kirk's monologue before the theme music is the one that's basically become the Star Trek theme, but the theme itself has only been used in TMP before the reboot, which is why I was pleasantly surprised to hear it over the end credits in 2009.
They're both part of the main title theme. The theme is the entire piece of music that played under the main titles, and it has two different parts, the "fanfare" under Shatner's narration and the other part, which I think of as the "song" portion (since it's the part Roddenberry's lyrics go with, and is structured like the popular songs of the day, specifically being a pastiche of "Beyond the Blue Horizon" with elements of "Begin the Beguine"). Courage used the song portion of the theme in TMP, the other movies and the TNG titles used the fanfare portion, and Giacchino's end titles use both. And if you look at the credits on the soundtrack albums, they're all attributed as "Contains 'Theme from STAR TREK' by Alexander Courage."