Anne Boleyn was indeed crowned with the crown of St Edward sitting in St Edward's Chair, the Coronation Chair - see https://www.theanneboleynfiles.com/1st-june-1533-the-whit-sunday-coronation-of-anne-boleyn/5625/ and p179 of "The Life and Death of Anne Boleyn" by Eric Ives.
I've looked in Letters and Papers and also in Hall's Chronicle and they make no mention of St Edward's Chair. Hall simply says that she sat in a "riche chayre" (p803), but Henry had her crowned with St Edward's Crown which was usually only used for crowning the actual monarch, not his consort, so I'm sure she would also have used St Edward's Chair. Henry was making a point that she was Queen and should be accepted as such.
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As an interesting side note: Katherine of Aragaon and Anne Boleyn were the ONLY wives of Henry VIII that enjoyed a formal coronation. It’s been suggested that by the time Jane Seymour came along, it was far too expensive a venture, since it on for days before and after the actual ceremony. The whole of London, and its visitors, would have been fed and entertained (to some extent) by the King’s treasury. By 1536, Henry’s marriage to Jane, Henry was far too involved in bribing the nobility with the lands and possessions of the monasteries and other religious houses to waste precious capital on a massive party! He had consciences to bribe!
Some have also implied that Henry wanted to see if Jane could produce the goods, so to speak, before going through that expense and fuss.