While in his memoirs Grant claimed that he made the final decision, I think there is strong evidence that Rosecrans's fate was already determined by Stanton.
Indiana Governor Oliver P. Morton met with Stanton at the depot in Indianapolis, before Grant's train arrived. Morton would later claim that Stanton told him that Rosecrans was being replaced because Rosecrans had lost the will to fight. Stanton based this conclusion (very erroneously, or perhaps maliciously, on a telegram Rosecrans sent to Lincoln on about October 3rd, proposing a truce and amnesty offer to CSA soldiers.)
Stanton told Morton that Rosecrans had suggested a general amnesty and truce, further suggesting that Rosecrans thought the war was unwinnable. That is of course not at all the content of the original telegram, but rather Stanton's "spin" on the matter.
In short, I suspect that Grant received those two orders as a way for the government to have political cover, but in reality, Grant only had one real choice. Stanton was unlikely to be satisfied with Rosecrans's retention.
Morton's story appeared in an Indianapolis newspaper in the 1870s. Lamers cites it as well.