You may find this post where /u/Libertat and I discusses contemporary naming conventions for the Early Medieval English useful.

The tl;dr of that is that "Anglo-Saxon" isn't really a term that was used at the time by the English to describe themselves. As /u/Libertat discusses, the term "Angle" was often used by external sources to talk about the English, and may well have contained an element of Christian identity, but wasn't really a term used by the "Angles" themselves, at least in the Early Anglo-Saxon Period. Bede is responsible for the model which has survived until very recently and still lingers on of lowland Britain being carved up into neat little kingdoms of Angles (Mercia, Northumbria, East Anglia), Jutes (Kent) and Saxons (Wessex, Essex, Sussex etc.) but in surviving documents, those "Anglian" and "Jutish" kings tend to identify themselves by kingdom (e.g. Rex Merciorum, Cynges Cantwaras) rather than as Angles, while the rulers of Wessex did to identify as rex Saxonum.

"Anglo-Saxon" appears very briefly in the period, in a brief period shortly before English unification in which West Saxon kings Alfred and Edward use Rex Anglorum et Saxonum - occasionally contracted to Rex Angolsaxonum - to emphasise their control over both Saxon heartlands and wider "English" territories. The idea of a pan-English identity was a useful political tool for the 'Cerdicing dynasty as it attempted to forge a new kingdom based as much on occupying historical rivals as it was 'liberating' them from the Danes. Broadly speaking, however, Angolsaxonum was replaced very quickly by the simple idea of the Angelcynn or gens Anglorum.

"Anglo-Saxon" as a common term really emerges in the 16th century with antiquarians such as William Lambarde, and then really explodes into popularity in the 18th and 19th centuries, where it becomes a rhetorical short-hand for "English between 410 and 1066". As I'm sure you can imagine, 18th and 19th Century 'histories' of the "Anglo-Saxons" in particular usually sought to portray the English as a distinct racial group, free from European (read Catholic) influence and naturally superior to other 'British ethnicities'. Parker's England's Darling: The Victorian Cult of Alfred the Great is a really good look at how Early English history was used, manipulated and even invented largely in the interests of Imperial British (English) propaganda.