The title senior administration official is a term used by the American press to indicate the identity of a source while retaining his or her anonymity.
As the title is subjective, the reporter writing the article is allowed to decide if a source should be called one.
However, senior administration officials almost always have the rank of Assistant.
Other people that can be classified using this title include the Vice President and Cabinet secretaries (occasionally deputies and undersecretaries as well).
Sometimes officials request that they be identified using other titles to prevent anybody from determining their true identity.