The Japanese surrender offer of August 10th that was passed to the Americans was that:

The Japanese Government are ready to accept the terms enumerated in the joint declaration which was issued at Potsdam on July 26, 1945, by the heads of the Governments of the United States, Great Britain, and China, and later subscribed by the Soviet Government with the understanding that the said declaration does not comprise any demand which prejudices the prerogatives of His Majesty as a Sovereign Ruler.

It's that "with the understanding" part that has the condition in it — what does "not comprise any demand which prejudices the prerogatives of His Majesty as a Sovereign Ruler" mean?

The Americans has a sense of what it meant because they had been monitoring Japanese foreign communications (the MAGIC decrypts) for some time, and understood that the Japanese were attempting to preserve their imperial system (the kokutai), which had been a sticking point with earlier deliberations about surrendering within the Japanese Supreme War Council. This issue had been understood and anticipated prior to the release of the Potsdam Declaration, and Truman had deliberately not clarified the US position on this.

But in any event: the Americans rejected this, and replied that:

"With regard to the Japanese Government's message accepting the terms of the Potsdam proclamation but containing the statement, 'with the understanding that the said declaration does not comprise any demand which prejudices the prerogatives of His Majesty as a sovereign ruler,' our position is as follows:

"From the moment of surrender the authority of the Emperor and the Japanese Government to rule the state shall be subject to the Supreme Commander of the Allied powers who will take such steps as he deems proper to effectuate the surrender terms.

"The Emperor will be required to authorize and ensure the signature by the Government of Japan and the Japanese Imperial General Headquarters of the surrender terms necessary to carry out the provisions of the Potsdam Declaration, and shall issue his commands to all the Japanese military, naval and air authorities and to all the forces under their control wherever located to cease active operations and to surrender their arms, and to issue such other orders as the Supreme Commander may require to give effect to the surrender terms.

"Immediately upon the surrender the Japanese Government shall transport prisoners of war and civilian internees to places of safety, as directed, where they can quickly be placed aboard Allied transports.

"The ultimate form of government of Japan shall, in accordance with the Potsdam Declaration, be established by the freely expressed will of the Japanese people.

"The armed forces of the Allied Powers will remain in Japan until the purposes set forth in the Potsdam Declaration are achieved."

Which was what the Japanese eventually accepted on August 14th without qualification.

It sounds like you are confusing two things: Japan's specific August 10th "conditional" surrender offer quoted above, and the Japanese "peace party's" earlier considerations about what kind of surrender they might be willing to negotiate prior to the atomic bombs/Soviet invasion. The latter were never finalized or offered up so we do not know what they might have asked for; we only know that they were considering ways of ending the war and what terms they might find acceptable. At the core was always the preservation of the kokutai, but there were some who (delusionally) believed that they might press for more once they were at the negotiating table. None of the latter came to pass, in part because the imagined mediator of this negotiation — the then-neutral Soviet Union — rejected such overtures and declared war on the Japanese.

For an in-depth treatment of these events, see Hasegawa, Racing the Enemy.