The immediate answer to the question posed in the title is given with characteristic dry clarity by James Murray in that great work of English history the Oxford English Dictionary. Murray's first definition is “English Saxon, Saxon of England: orig. a collective name for the Saxons of Britain as distinct from the ‘Old Saxons’ of the continent. Hence, properly applied to the Saxons (or Wessex, Essex, Middlesex, Sussex, and perhaps Kent), as distinct from the Angles.” After explaining that, “in this Dictionary, the language of England before 1100 is called, as a whole, ‘Old English,’”Murray then goes on to say that the adjective “Anglo-Saxon” is “extended to the entire Old English people and language before the Norman Conquest.” Neither he nor the Supplement mentions explicitly the almost purely chronological use of “Anglo-Saxon” to describe the whole period of English history between 400 and 1066 that is now current, but it is easy to see how this has derived from the usage they expound.
What the original edition goes on to do, moreover, is to give an account of a wider use of the word that beautifully encapsulates the beliefs about culture and descent that lie behind it. The expression “Anglo-Saxon,” according to the Oxford English Dictionary, was then—that is, in the late nineteenth century—used “rhetorically for English in its wider or ethnological sense, in order to avoid the later historical restriction of ‘English’ as distinct from Scotch, or the modern political restriction of ‘English’ as opposed to American of the United States; thus applied to (1) all persons of Teutonic descent (or who reckon themselves such) in Britain, whether of English, Scotch, or Irish birth; (2) all of this descent in the world, whether subjects of Great Britain or of the United States.”