There were additional terms that Anami and the war faction wanted included if they were to agree to surrender, which you have listed. However, these were never communicated to the Allies. For the most part, the answer you linked covers all communication of importance between Japan and the belligerent Allies.

The demand for the "unconditional surrender of Germany and Japan" was originally announced by FDR at Casablanca in January 1943 much to the chagrin of Churchill and Secretary of State, Cordell Hull (who FDR never really included in anything of importance.)

The Potsdam Proclamation, which was issued by the US, Britain and China on July 26, 1945 stated:

We call upon the government of Japan to proclaim now the unconditional surrender of all Japanese armed forces, and to provide proper and adequate assurances of their good faith in such action. The alternative for Japan is prompt and utter destruction.

The changing of language from "unconditional surrender of Japan" to "the unconditional surrender of all Japanese armed forces" was significant as this did exclude the Emperor from unconditional surrender. This was recognized as a significant concession by the Japanese Foreign Ministry and Foreign Minister Togo, but Togo also knew it would not budge Anami and the war faction. There were also other conditions that were included in the Proclamation, such as that the military would be allowed to return to their homes, and Japan could maintain such industries as would sustain its economy. US Undersecretary of State, Joseph Grew, wanted to add some additional language to the Potsdam Proclamation about retention of the Imperial Dynasty, but I'll get to that later.

Japan's first reply other than Domei radio broadcasts which were controlled by the military was on August 10th, to accept Potsdam with the caveat that, "the said declaration does not comprise any demand which prejudices the prerogatives of His Majesty as a Sovereign Ruler."

The US replied after conferring with Churchill, Stalin and Chiang on the 11th, with concurrence with the exception that, "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."

There were high level leaders such as Anami within the Japanese government who wanted to include other conditions, but these were never officially communicated to the Allies. The fact that the government was required to reach a unanimous decision, and was split led to the ability of the Emperor to decide for them, and accept surrender. The legal authority to surrender actually lay with the cabinet, which did not include the Chiefs of Staff Toyoda and Umezu, although they would have likely rubber stamped any decision by the Big Six. The Home Minister and Justice Minister also sided with Anami.

Regarding Joseph Grew's attempt to include language to retain the Imperial Dynasty, there are a lot of common beliefs that don't agree with what Joseph Grew had to say on the matter. Often that was portrayed as retention of Emperor Hirohito. Grew on the other hand said that he wanted to keep "the Imperial polity under the current dynasty." He desired to wait until they reached Tokyo to determine whether that would mean Hirohito or another arrangement, likely Prince Akihito under some sort of regency.

Byrnes is often portrayed as the one who kept the language from being included in the Potsdam Proclamation, but he took the job of Secretary of State just a few days before leaving for Potsdam, and Grew was barely able to shove the draft language into his pocket on the way to the airport. Grew had taken the suggestion to Truman a month earlier. While Truman was amenable, he wanted Grew to get concurrence from the Chiefs of Staff. At that meeting on May 29, Stimson, Forrestal, and Marshall (King being absent) liked the idea, but felt the timing was not right as they were currently in heavy fighting on Okinawa and would convey weakness. As to whether the inclusion would have ended the war sooner, Grew wrote:

That question can probably never be definitively answered but a good deal of evidence is available to shed light on it. From statements made by a number of the moderate former Japanese leaders to responsible Americans after the American occupation, it is quite clear that the civilian advisors to the Emperor were working toward surrender long before the Potsdam Proclamation, even indeed before my talk with the President on May 28, for they knew then that Japan was a defeated nation. The stumbling block that they had to overcome was the complete dominance of the Japanese Army over the government, and even when the moderates finally succeeded in getting a decision by the controlling element of the Government to accept the Potsdam terms, efforts were made by the unreconciled elements in the Japanese Army to bring about the nullification of that decision. The Emperor needed all the support he could get, and in the light of available evidence I myself and others felt and still feel that if such a categorical statement about the dynasty had been issued in May, 1945, the surrender-minded elements in the Government might well have been afforded by such a statement a valid reason and the necessary strength to come to an early clear-cut decision.

But we will never know if that would have played out that way or was just wishful thinking on Grew's part.

Sources:

Grew, Joseph C. Turbulent Era

Togo, Shigenori The Cause of Japan

The Pacific War Research Society Japan's Longest Day