It's interesting, the way I've heard the story about Thad's rendition of the quote on the Basie April in Paris session is that the band had made many takes on the tune, and in Thad's efforts to avoid repeating himself in subsequent takes he played the quote on the take which ended up being the "keeper". The downside was that when the band performed the tune thereafter Thad was obligated to repeat the quote, since it was part of the popular recorded version of the tune with which audiences were familiar. You still hear players dutifully playing the quote (most of the time they don't even realize it was Thad's), recreating the famous solo (similar to the recreation of Bobby Hackett's "String of Pearls" trumpet solo).
This has always been a difficult reconciliation for jazz musicians- the desire to be spontaneous and unique on every performance which conflicts with the desire of the audience to hear something familiar. It happened to me once on Woody's band when I played on Bijou, a piece made famous by Bill Harris. After the set, someone was upset I didn't play the "famous Bill Harris solo". My response was that even Bill Harris didn't play the "famous Bill Harris solo" every night. It's that concept of the repertory band- I believe in studying the recording and attempting to play in the appropriate style, but playing someone else's EXACT solo is (IMHO) impossible to do with anything approaching the same level of spontaneity. It's like when someone says "tell that funny story"- it almost always bombs because it's too contrived and not at all spontaneous.