Perhaps one of them was the flag of the Alliance as it existed on Earth (i.e. the actual union of the USA and China) and the other one was the flag of the space based organization known as the Union of Allied Planets?
Grasping at straws, yes. But it would also explain the different names (i.e. "Anglo-Sino Allliance" and "Union of Allied Planets"). Who says they have to be the same organization? The former may have been Earth only, whereas the latter is the one in use currently.