You are right. Looking at the Jaffe–Quinn paper again, it is closer to the distinction that Eliezer was making. (However, I note that the mathematical "theoreticians" in that article are generally high-prestige, and the "rigorous mathematicians" have to fight the perception that they are just filling in details to results already announced.)

My mischaracterization of Jaffe and Quinn's thesis happened because (1) Thurston replied to their article, and he discusses computer-aided proofs in his reply; and (2) even more embarrassingly, I conflated the Jaffe–Quinn article with the Scientific American article The Death of Proof, by John Horgan.