Yes, I think we generally agree here. What seems clear from archived information from Japan is that its leaders were not concerned with another bomb (even if it was nuclear.) By that point, 68 cities in Japan had been bombed by US conventional air raids. The destruction was incredible, some of the most destructive bombing of individual cities history had seen up to that point. However, it wasn't until the USSR broke their neutrality pact and invaded Manchuria that the leaders met and began discussing unconditional surrender. For some time, Japan had entertained the idea of having Stalin act as a moderator for negotiations between Japan and the US. Once the Soviets entered into war, that option was gone. Japan was also prepared to defend an invasion from the Allies from the south, but knew they could not defend from both the Soviets and the US on two fronts. This appears to be what caused Japans leaders to meet and discuss unconditional surrender.

Stalin apparently suggested to Truman that they would invade Manchuria on August 15th. This was known before the US dropped the bombs on Hiroshima and Nagasaki. There is some evidence that the US, on hearing this news, anticipated this would lead to Japans unconditional surrender. The US had a planned ground invasion (Operation Downfall) but it was slated for November. My understanding is they were having a hard time getting material support from the UK, France, etc. because of how much longer they had been in the war. From Truman's perspective, conventional bombing was making zero progress towards ending the war with Japan, as its leaders seemed content to let it continue.

As it became more clear over time that Japans situation was deteriorating, it became more obvious that Russia entering into war with Japan was to cause them to surrender. There is evidence that over time, this changed the dialog internally from strategic and tactical use cases for the bomb to political use cases. They believed that use of the bomb would end the war before Russia made much headway in Manchuria. They believed that using the bomb would ensure they do not have to share victory over Japan with Russia. Furthermore, they thought the bomb was a way to strengthen America's diplomatic hand not only in the Far East but in negotiations over the fate of Europe in general, and Eastern Europe in particular. James Byrnes' view on the matter was that possessing and demonstrating the bomb would make Russia “more manageable” in Europe.

The bomb was ultimately dropped on August 6th, The Soviets invaded Manchurian on midnight August 9th, and the second bomb hit Japan later that morning. It's clear that the bomb dwarfed the significance of the Soviets invasion, at least in the public eye and the eye of American history, but it wasn't lost on the leaders of Japan.

You're definitely right about concerned surrounding war crimes. The trials were already underway regarding Germany when the bombs were dropped. I'm not sure if they saw the US as a more sympathetic power, however, it seems clear that Japanese leaders utilized the spectacular nature of the two atomic bombs as a cover for their failings. This cover worked twofold, it allowed them to save face with the Japanese public, who had very little information about the effectiveness of Japans war generally, while also stroking the Americans ego. If the specular shock and awe of this unforeseen weapon of mass destruction was the origins of unconditional surrender, then it wasn't because of their poor handling of the war. The Japanese leadership utilized the horrors of these bombs to obscure the horrors they were guilty of. This ultimately strengthened the US's position as having redefined the nature of war, giving them what they desired: More influence over post-war Japan, and influence across Europe as a whole.