To add to this, various Norwegian and Danish viking groups inhabited Eastern Britain for some time, why are they omitted all together?

I wager it had something to do with Alfred the Great being the King of the West Saxons during his defense of England against the Vikings. After his victories, he styled himself 'King of the Anglo-Saxons' (sorry, Jutes/Frisians) but was still himself Saxon. So the language and country takes it name from the Angles (England), but the monarchy that ruled was one of a Saxon king. They are Anglo-Saxons of a Saxon Kingdom.

There were different tribes. The reason the island was called "Engla lond," "Land of the Angles," comes down King Alfred the Great's need to unify the diverse kingdoms that made up modern day England in the face of the Viking threat. Scholars see his literary program as a cultural bulwark against the Vikings who were ravaging England's highly advanced centers of learning. Alfred needed to create a sense of a unified English identity to do this. Part of this program was the creation of the Anglo-Saxon Chronicle, which lays out the origins of a unified AS people on the island, underscoring their claim to the island and Alfred's claim to be the king of all England.

For this, see

Parkes, MB. "The Paleography of the Parker Manuscript of the Chronicle, Laws and Sedulius, and Historiography"

Pratt, David. The Political Thought of King Alfred the Great

What I wrote above is a brief thumbnail sketch. In essence, the very idea of England is something that goes back at least to Bede. You can probably trace it further back to Gildas and Caesar before him (two authors that Bede used to structure his account of the Historia ecclesiastica gentis Anglorum, The Ecclesiastical History of the English People.)

Why were the Angles chosen over the Saxons? There are many reasons that are geopolitical and relate to who was in power when the name was coined. Part of it is literary however: There is a famous scene that Pope Gregory wrote that was recounted by Alfred, in which he puns on Angles and Angels in a Latin text that describes his motivation in sending St Augustine to the island to convert the Germanic peoples there. It is partly on the strength of that pun that the Angles were the ones chosen for the name of the island.

The Anglo-Saxon past has been an important cultural site for the English since Old English texts were rediscovered by the Elizabethans, who were mining their past to find justification for their new Anglican church. They found it in Old English religious writing. The AS past continued to be extremely important, and this reached its highest point in the Victorian era. This was an important time both for scholarship and for England's culture, and the Anglo-Saxon past was more accessible than ever just when the English were considering who they were racially and socially. This was likely spurred by experiences in other societies and cultures where they had an imperial presence. If you read Carlyle, Tennyson, Scott, etc., you can see the Victorians working out who they were by looking to their past.

For more on this see:

Brantlinger, Patrick. Victoran Literature and Postcolonial Studies.

As for why the Victorians, who popularized this, referred to the Saxons more than the Angles, despite the fact that they knew it was called Englalond, I can personally only speculate. My speculation is that Saxon is sufficiently different and alien than Angles would be. Saxon hearkens back to the old continental past that the Victorians wanted to connect to and explore. This is something that was also true of the Anglo-Saxons themselves: they wanted to connect to their Saxon past as well, and made special attempts to convert the Saxons who they felt shared ancestors with them.

For more on this, see

Yorke, Barbara. "The Insular Background to Boniface's Continental Career".

Also, the Anglo-Saxon Christianity and Saint Boniface pages on Wikipedia have very very good bibliographies.

I'd write more, but I have to go. Sorry if this seems cut short; I can go on and on about this, but you have to stop some time!

The funny thing is that England was known by several nations by the Saxons, not the Angles. The Scots (Gaelic speakers) call England Sasainn. England in Irish Gaelic is Sasana. Both of these are terms for Saxon.

There have been some excellent answers to this question already but this might help to fill in some of the detail. Various kingdoms were set up by the Angles and Saxons, Sussex, Essex, Wessex, Mercia, Northumbria, Anglia, Hwicce are some of them. They eventually coalesced into larger kingdoms, including Northumbria, Mercia, East Anglia and Wessex. The vikings (mostly Danes) then arrived and destroyed all of the kingdoms except for Wessex, the West Saxons. The kingdom of Wessex was able to defeat the Danes and eventually expanded to become a single kingdom that ruled all of the Anglo-Saxon territories. Hence, although the majority of the population could be described as Angles, and the country as Angle-land (I've modernised all spellings) the king was the king of the West Saxons, so could be described as a Saxon, and the people of the Saxon king could, and would, be described as Saxons.