The tricky part about answering this question is that the entirety of the "decision-making process" for the use of the atomic bomb (in which Truman barely played any part) was about the use of the first atomic bomb. In retrospect this looks bizarre to us, because we put so much weight on there being two bombs, but the predominant thinking at the time was that the bombs would not immediately end the war, and the total number of atomic bombs that might need to be dropped could be many. So it was less about "two bombs" than "the first bomb, and then more would follow." The authorization order (which Truman saw, but was only explicitly approved by the Secretary of War) said, after giving specifics on the first use of the bomb: "Additional bombs will be delivered on the above targets as soon as made ready by the project staff. Further instructions will be issued concerning targets other than those listed above." Which was interpreted by General Groves (who wrote said order) as giving him a totally free hand to determine how any subsequent bombs would be used after the first.

Truman was not involved in this. It is not clear to me that he even knew there were more than one bomb ready in early August 1945. He was very peripheral to all of this process, and he seems to have only conceived of the atomic bomb as a singular thing.

The military officials, like Groves, had always assumed that the two bombs would be used, though initially there was a week between them in their schedule. As it was, weather conditions pushed Hiroshima later and the second bomb earlier, so there were only three days between them. The exact timing decision was made by the people on Tinian, and relayed to Groves.

(And Nagasaki was not the intended target for the second bomb; Kokura was. Nagasaki was a target added at the last minute to the strike order, because they had removed Kyoto from the list and needed another backup target near Hiroshima and Kokura. Clouds over Kokura led to its being spared.)

Anyway. The long and short of it is, "nobody really made an explicit decision" — like much of the atomic bomb use during WWII it was less about deliberation and more about a train set into motion that picked up a momentum of its own. But if one were going to pick a single person to credit it to, I'd go with Groves — he set much of that into motion. Definitely not Truman — not only did he not take part in any decision about using a second bomb, again, I'm not sure he even knew one was going to be used.

I find the Nagasaki bombing as a whole very interesting, because it contradicts a lot of the received narratives about the use of the atomic bombs, including the centricity of Truman's "decision to use the bomb" (which is a myth). I've written on Nagasaki here, for what that is worth.

Alex Wellerstein is among the principle authorities on this subject, and I don't think any of his work names names.

His writing, most of which is made publicly accessible through his blog,* persistently insinuates that Truman didn't "order" the dropping of the bomb, but the semantics of what that means is not quite clear. Wellerstein sees Truman as someone who felt it was important to claim responsibility for the bomb (which is not the same as believing the Truman ordered it), but Wellerstein also argues that it is after Nagasaki that Truman decides that the power over an atomic bomb ought to be a presidential decision, not a military one (the 1946 Atomic Energy Act). I always feel like I am reading between the lines with Wellerstein.

Wellerstein's article from the New Yorker may also be of interest as might this one from the Washington Post.

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* While "anyone" with a blog isn't a legitimate source, Wellerstein's blog is considered an academically legitimate piece of public facing history.