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It's been years since I read the Jaffe–Quinn article. But, as I recall, it was more about the methods used to answer questions, and about how rigorous human-verifiable proofs might give way to heuristic/probabilistic and computer-aided proofs. Eliezer, on the other hand, seemed to be saying that mathematicians concentrate prestige on answering questions (by whatever means the community considers to be adequate), as opposed to "figuring out the shape of the right theorem to be proved".
Jaffe and Quinn mainly advocate that labor should be divided between people who make conjectures ("theoreticians") and people who prove them ("experimentalists"). I don't think there is much of anything about probabilistic or computer-aided proofs.
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.