And by that I mean why does the Norman invasion mark the beginning of how it’s basically described and recorded. For instance Edward I is Edward I despite previous Edwards being king. What was it about William I that led to this shift?
Mentioning a previous answer is not intended to discourage more discussion, questions, information, or debate.
For regnal numbering,
/u/Miles_Sine_Castrum wrote about it in "Why is it that the regnal numbers of the monarchs of England start from the Norman Conquest? For example, why is Edward I not known as Edward II and Edward the confessor known as Edward I?", https://www.reddit.com/r/AskHistorians/comments/5g3cjx/why_is_it_that_the_regnal_numbers_of_the_monarchs/
/u/Valkine wrote about it in "If Arthur, Prince of Wales had succeeded his father Henry VII, would he have been King Arthur I or II?" at https://www.reddit.com/r/AskHistorians/comments/6po4f3/if_arthur_prince_of_wales_had_succeeded_his/
This is a difficult question to answer without knowing the specifics of what you mean by "English." You could be asking when did the history of England as a unified country begin. You could be asking when did the English begin regarding themselves as a people separate from other Germanic tribes. You could be asking something more nebulous, for example, when did the idea of a unified "English history" begin, regardless of whether England itself was unified. Either way, the answer is nuanced. Nevertheless, English history certainly began well before the Norman Conquest in 1066.
Cnut was the first King to style himself as "King of England" in 1016. He was a Dane, and did not style himself as "King of the English" as his predecessors had. However, England itself was united well before Cnut, and English history definitely began prior to his rule.
Athelstan was the first king to style himself as "King of the English" in 927 upon conquering York, the last remaining Viking kingdom, making him the first Anglo-Saxon king to rule all of England. I would argue this is the beginning of England as a united country. However, the idea of unified England is usually attributed to his grandfather, Alfred the Great. Prior to 927, kings styled themselves as "King of the Anglo-Saxons". "English" and "Anglo-Saxon" are synonymous (I would love to be corrected if I am wrong about this). Alfred the Great was the first king to style himself as "King of the Anglo-Saxons," and he first did so when he defeated the Vikings in the Battle of Edington in 878, setting the borders of England ruled by the Anglo-Saxons and England controlled by the Danelaw (a group of independent Viking territories subject to Danish law). In a nutshell, England was unified in 927, but an English identity and the prospects of a unified England were forged in 878.
Now, if you are asking about when does English history begin with regard to the English people, Alfred the Great was originally King of Wessex beginning in 871. Wessex was one of several Anglo-Saxon kingdoms existing prior to the Great Heathen Army (the Viking invasion of England in 865). The Anglo-Saxon kingdoms were established during the Anglo-Saxon settlement of Britain in the fifth century (which was composed of the Germanic tribes of the Angles, Saxons, and Jutes) following the withdrawal of Roman power from Britain around 410 AD. So, I would argue the history of the English people began with the migration of Angles, Saxons, and Jutes in the Fifth Century, and the supplementation and integration of Germanic culture with the Sub-Roman culture of native Britons.