Do you know what the difference is between an endonym and an exonym?
An exonym or xenonym is an external name for a geographical place, a group of people, an individual person, or a language or dialect1. It is a common name used only outside the place, group, or linguistic community in question. An endonym or autonym is an internal name for a geographical place, a group of people, or a language or dialect. It is a common name used only inside the place, group, or linguistic community in question; it is their name for themselves, self-designated, their homeland, or their language.
In the case of the Germanic invaders of Britain, "Saxons" seems to have been both an endonym of some of them and an exonym for all of them.
The Saxons (Latin: Saxones, German: Sachsen, Old English: Seaxan, Old Saxon: Sahson, Low German: Sassen, Dutch: Saksen) were a group of early Germanic1 peoples whose name was given in the early Middle Ages to a large country (Old Saxony, Latin: Saxonia) near the North Sea coast of what is now Germany.2 In the late Roman Empire, the name was used to refer to Germanic coastal raiders, and also as a word something like the later "Viking".3 Their origins appear to be mainly somewhere in or near the above-mentioned German North Sea coast where they are found later, in Carolingian times. In Merovingian times, continental Saxons had also been associated with the activity and settlements on the coast of what later became Normandy. Their precise origins are uncertain, and they are sometimes described as fighting inland, coming into conflict with the Franks and Thuringians. There is possibly a single classical reference to a smaller homeland of an early Saxon tribe, but its interpretation is disputed (see below). According to this proposal, the Saxons' earliest area of settlement is believed to have been Northern Albingia. This general area is close to the probable homeland of the Angles.4
In late Roman times Germanic tribes from roughly the area in and around north western Germany were called "Saxons", regardless of their specific ethic identity. So the Germanic sea raiders in Britain and Gaul who came from that area were all called Saxons.
So when members of various groups from that region invaded and/or settled in Britain, the Romano-Britons called them Saxons.
In modern times words derived from Saxon are used for English people in some languages.
The Celtic languages of the British Isles use terms derived from Old English Seaxan, 'Saxon', possibly itself derived from Old English seax:
Scottish Gaelic: Sasannach, in older literature Sacsannach / Sagsananch; the language is Beurla. Sassenach is still used by Scottish speakers of English and Scots to refer to English people, mostly negatively.
Cornish: Sows, plural Sowson; the English language is Sowsnek
Welsh: Sais, plural Saeson; the English language is Saesneg
Irish: Sasanach, historically also having the colloquial meaning "Protestant"; the language is Béarla, short for Sacs-Bhéarla "Saxon language"
Manx: Sostynagh, plural Sostynee; the English language is Baarle, from Irish
The "Byzantine" writer Procopius writing about 550, said that Britain was settled by Germanic tribes called Angles, Saxons, and Frisians.
The English monk Bede, writing in the first third of the 8th century, two hundred years after Procopius, said that Britain was settled by Germanic tribes called Angles, Saxons, and Jutes, and he named the regions and kingdoms settled by the various groups.
According to sources such as the History of Bede, after the invasion of Britannia, the Angles split up and founded the kingdoms of Northumbria, East Anglia, and Mercia. H.R. Loyn has observed in this context that "a sea voyage is perilous to tribal institutions", and the apparently tribe-based kingdoms were formed in England. Early times had two northern kingdoms (Bernicia and Deira) and two midland ones (Middle Anglia and Mercia), which had by the seventh century resolved themselves into two Angle kingdoms, viz., Northumbria and Mercia
The Angles apparently came from Anglia or Angeln and other parts of southern Jutland.
The Jutes, along with some Angles, Saxons, and Frisians, sailed across the North Sea to raid and eventually invade Roman Britain, from the late fourth century onwards, either displacing, absorbing, or destroying Romanised British kingdoms in south-east Britain. According to Bede, the Jutes established four kingdoms:
Cantaware (Latinised as Cantuarii), in the Roman civitas of Cantiaca (Kent);
the kingdom of Wihtwara (Latin: Uictuarii) on isla Vectum (the Isle of Wight)
two settlements in coastal parts of the previous kingdom of the Belgae, i.e. an area known later as Hampshire:
Meonwara (in the Meon Valley area), and
Ytene (which Florence of Worcester states was in the area that became the New Forest).
Some evidence indicates that the Haestingas people who settled in the Hastings area of Sussex, in the 6th century, may also have been Jutish in origin.
According to Bede, the Saxons settled other kingdoms such as Essex, Sussex, and Wessex.
If you look at a map of Anglo-Saxon Kingdoms, you will see that Northumbria, Mercia, and East Anglia covered about two thirds of England, while Wessex, Sussex, and Essex covered about two thirds of the rest of England, and the Jutish kingdoms covered about a nineth of Anglo-Saxon Britain.
Thus it seems fairly reasonable for southern Britain to be known as England instead of Saxonland.
England was united by the Kings of Wessex, whose people were mostly of Saxon ancestry.
I believe that Alfred the Great and his son Edward the Elder usually used the title of Rex Anglosaxorum "King of the Anglo-Saxons", King Aethelstan I is said to have used the title of of King of the Anglo-Saxons" from 924 to 927 and the title of Rex Anglorum "King of the English" from 927 to 939. It was only natural for kings of Saxon ancestry to try to make Saxons seem equally as important as Angles in their title. But they seem to have given that up in the time of Aethelstan I.
After the Norman conquest, the title was King of the English from 1066 to 1154, and King of England from 1154 onward.
[Added May 31, 2020. I note that Bede, who was an Angle, wrote somewhere that Angle was not only the name of the most numerous tribe of Germanics in Britain, but also the generic term of all the Germanics in Britain. If he wasn't just bragging, even the Saxons and the Jutes would have sometimes used Angles as their endonym, while apparently Jutes and Angles never used Saxons as their endonym.
I note that northern and eastern England, with their populations of Angles, were ruled by Dane since the time of Alfred the Great, while Aethelstan was the first king to rule all those lands as well. So he may have chosen the title of King of the English to emphasized that he now ruled all of the Angles.]