The tricky part of the question is whose motivation are you asking about? Are you asking about the scientists, the military men, the civilian advisors, or Truman himself? Because for each of these categories, much less individual people within these categories, there were sometimes different motivations and multiple motivations. Human beings, as you have no doubt experienced, frequently have more than one reason for doing or wanting things.

So for General Leslie Groves, the head of the bomb project and the architect of much of its planning and policy (who is often overlooked in just looking at Truman), it is clear that he had an overriding motivation to use the bomb somehow before the war ended. Why? In part because he did want to contribute to the war. But also because he had sunk his reputation into this project, and believed that should he fail to show results for it, he would spend the rest of his life before Congressional hearings on wasted resources. Did he want to use the bombs to hasten the end of the war? Sure, of course. But here's the rub: he did not believe that one or two atomic bombs would end the war. He believed it might take many more bombs, plus the threat of invasion, etc., to end the war. So the "he knew if two bombs were dropped the war would end" narrative is a false one: neither he nor anyone else actually expected the bombs to be credited with ending the war immediately, and was actually surprised when the Japanese surrendered.

(There is a separate historical question of what role the bombs played or did not play in ending the war. We don't get into that here, for the sake of space, but I just want to point out that this is itself contested historical territory.)

The scientists who built the bomb were largely too focused on technical success to question exactly what their long-term motives were. Most had started the project because they were afraid of a German bomb, but very few questioned whether the defeat of Germany, and the transition of the weapon from a deterrent to a "first strike," made any difference. (After the war they would question this quite a bit.) Some asked whether the bomb should be "demonstrated" before used for mass slaughter. Some of them, such a the scientific head J. Robert Oppenheimer, had a very peculiar motivation to wanting to use the bomb on a city: they were looking ahead to the postwar and Cold War, and believed that if the first use of the atomic bomb was not sufficiently spectacular then the political implications of a weapon that, in its more evolved forms that they could also foresee, was a nation-killer, would not be understood. So for Oppenheimer, one major motivation was the belief that unless it was made very clear how terrible nuclear weapons were, the world would plunge into an arms race that would make human extinction possible. His alternative was a plan for what was called "international control," which would basically be a scheme in which the United Nations would ban nuclear weapons and, perhaps, war itself. So that's a very different sort of motivation, one easily overlooked, despite the fact that Oppenheimer was very much involved with setting and keeping in motion the plans to use it on a city.

So what of Truman and his advisors, who are usually the most focused on for this question? Their motivations were multiple. Did they hope that the bomb might end the war quickly? They did, of course. That was one reason for using it at least once. Did they also see that using the bomb would also have implications for the Cold War, and for showing power to the Soviet Union? Some of them explicitly expressed this sentiment, notably Secretary of State James Byrnes. He expressed several times that he thought it would make Russia "more manageable" for them to know that the US had such military power. Was that his only motivation? No. But it was in the pile of motivations.

And Truman himself? He never ordered in any assertive way that the bomb be used. The plans were set in motion separate from him. He was aware of them to various degrees of awareness (he knew very much about the first bomb being used; it is not clear he knew that a second would be used so quickly). The one place we see an inkling of motivation is in how he handled the knowledge of the bomb at the Potsdam Conference: he seemed very concerned with the bomb being used prior to the forthcoming Soviet declaration of war against Japan, which he also seems to have believed would end the war prior to an invasion by the United States. In other words, he hoped the bomb would end the war quickly, not to avoid US invasion, but to avoid Soviet invasion, because the latter would give the Soviets considerable territory in the far east. Was that Truman's only motivation in using the bomb? Again, probably not. But was that in the mix? It appears to be the case.

As for the "saving of lives" argument: nobody made that argument prior to the use of the bomb. The "the bomb saved lives" argument was made only after the mass casualty reports from Hiroshima brought with it a wave of revulsion and outcry. Initially the lives claimed saved were Americans; only much later, with the end of the war, was it extended to claim that it had saved Japanese lives. And the number of lives supposedly saved has crept upwards and upwards over the years by people claiming the bomb saved them, from tens of thousands to hundreds of thousands to millions. It should be noted that the "it was all done to end the war and save lives and was very well-motivated and well-considered" version of the story only emerged in its full form in 1947, several years after the events itself, in response to growing criticism and argument that the bombs were unnecessary (including arguments from military generals, like Eisenhower). Which is only to point out that none of the arguments here are "neutral": if the argument that the Soviet Union was the primary motivation is "revisionist" (its first expression is from 1949, so it is not a significantly more "new" argument than the other one), then the argument that is now more "orthodox" (the "saved lives" argument) could easily be called "propaganda." All of these arguments were created to serve political ends, and historians have been sifting through them ever since.

Because there is a debate over whether the first bomb played a significant role in Japanese high command decisionmaking at all, much less the one used on Nagasaki (which seems very much not to have), it is hard to argue whether it actually saved lives. (The counterargument to "the bomb ended the war" is "the Soviet invasion, on August 8/9, ended the war.") But I just want to point out that each of these rhetorical justifications have their own histories, as to the allegations that the bomb was or wasn't important, was or wasn't dropped for one motivation or another, and so on.

There are no "authoritative" and "simple to interpret" sources about what was actually going on inside the minds of these people. There never is, for anything complex. We have a mess of evidence from the time that points to lots of different things being on their minds, and lots of motivations. We have their memoirs and recollections later which are always suspect as they tend to be self-justifying and also to rationalize a process that was not especially rational (because life rarely is).

All of the sentiments I've indicated above are documentable from contemporary sources. You will note that almost all of them point towards the use of the bomb. There were a few people who had motivations to not use the bomb (the like scientists who argued for a demonstration); they were not especially well-represented opinions at "the top." Historians sometimes call the use of at least one atomic bomb "overdetermined," because there were so many reasons that the top planners saw to use it (along with the "institutional momentum" of them having started along this path for years prior to it being used, and under a different President at that), and so few reasons not to use it (humanitarian concerns, while not absent, were not especially compelling in wartime, and resentment about Pearl Harbor, Japanese treatment of POWs, and other acts of Japanese atrocity and aggression made the Americans especially uninterested in anything that looked like coddling). I do think there was historical contingency (the possibility of something else happening) in some of the details, e.g., when the bomb was used (before or after Soviet invasion), how it was used (against a city or a more "military"/less "civilian" target, or a demonstration), and how many were used (the use of the second bomb was a far less planned and orderly process, and I'm not sure Truman would have approved having it be used so soon after the first were he asked or informed about it, but he was not). There is more one can say about these things, but I'll leave it at this. (I am traveling today and this week, so if I do not immediately reply to comments, don't be put off.)

For a great overview of historians' arguments over the motivation and how they have evolved over the years, and the evidence for them, see J. Samuel Walker, "Recent Literature on Truman's Atomic Bomb Decision: A Search for Middle Ground" Diplomatic History 29, no. 2, (2005), 311–334.