Yes, once granted refugee status an asylum seeker becomes officially a refugee. But in US immigration law, the reverse is not the case. A person seeking refugee status initially through the mechanism of immigration, which is how the classification "refugee" is defined, CANNOT by definition be an asylum seeker in the way such a status is defined legally. And that is true even if international standards recognize that the thing a refugee seeks is asylum, and even if the ultimate goal of asylum seeking is to be classified as a refugee.
The process of becoming an official refugee is bound by rules and regulations. To be classified a refugee one must have passed through that process. The designation of "asylum seeker" is a process by which one can obtain the benefits of a refugee before having met the legal requirements, and at least in theory without having to meet all the specific requirements (which is why the law is specific that asylum is not dependent on the country of origin or the expressed policy of the United States).
Of course the two things are nearly equivalent, and the desired result is the same, but there is an important and fundamental legal distinction here, which seems to elude you. Throughout this thread you have made the point that the law, opinion and moral stance of the international community is trumped by the laws and policies of the United States. Here we are with a case in which the law of the United States is clear as to the difference, but you seem ready to deny it because of what Amnesty International says.