Arthur Miller started The Crucible with a "Note on Historical Accuracy" because he didn't want people trying to learn the history of the Salem Witch Trials to read his play and assume it to be a true representation of events. While Miller used the names of real people in his play and some of the characters met the same fates in the play and in life, the personalities and motives of the characters in the play were crafted and created by Miller.
In the play, The Crucible, why would Arthur Miller include the Note on Historical Accuracy?
Here is the "Note on Historical Accuracy":
This play is not history in the sense in which the word is used by the academic historian. Dramatic purposes have sometimes required many characters to be fused into one; the number of girls involved in the 'crying out' has been reduced; Abigail's age has been raised; while there were several judges of almost equal authority, I have symbolized them all in Hathorne and Danforth. However, I believe that the reader will discover here the essential nature of one of the strangest and most awful chapters in human history. The fate of each character is exactly that of his historical model, and there is no one in the drama who did not play a similar — and in some cases exactly the same — role in history.
As for the characters of the persons, little is known about most of them except what may be surmised from a few letters, the trial record, certain broadsides written at the time, and references to their conduct in sources of varying reliability. They may therefore be taken as creations of my own, drawn to the best of my ability in conformity with their known behavior, except as indicated in the commentary I have written for this text.
Arthur Miller wanted to make it clear that he wasn't trying to write (or rewrite) history. This would have been impossible for him, given the lack of documentation about some of the key players of the Salem Witch Trial. Instead, Miller developed his characters to fit the play — which at the time it was published, was considered to be a statement about McCarthyism more than it was a retelling of the Salem Witch Trials.
This kind of historical fiction relies on what is called creative license. You'll see it used frequently in literature and film. (Sorry, but no couple named Jack and Rose cast off on the Titanic, and President Lincoln wasn't really a vampire hunter.)