It appears to me that both videos posted are from the same 1959 Berle Kraft Music Hall show [same set; same clothes, same camera angles, stool, etc.]. That show, like most 50s-60s variety shows, was done live. Note that Nick adds "true" before "story" to the intro in the Berle video which would point more to a live performance. While that word is not in the 1st video, there appear to be several edits from the Berle video including one at that point. So,
IF both are from the same performance, there is at least circumstantial evidence it was done live.
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John.....
I think you're right.
I haven't listened to the album version but....
The audio is awfully clear....
Notice that Nick has a felt pick on his headstock held by a rubber band...
There's this one.... same felt pick on Nick's headstock...I think they're the same