I am not sure what the claim in the textbook means, and have been unable to find any other reference to support or explain it. Headnotes are always “presumptively correct” in the sense that they are professionally authored, so you can generally presume them to be correct, unless they conflict with the full text of the decision. But I’m not aware of any statute or case law that creates a formal presumption in relation to headnotes. Perhaps you could write to the author and ask if that’s what she means.
What is unique about England and Wales, as the origin of the common law and the associated tradition of law reporting, is the preeminence of the Incorporated Council of Law Reporting. The ICLR was established in 1865 and publishes all official law reports in England and Wales. Arguably unlike the official or authorised reports of other common law jurisdictions, the ICLR’s reports are considered to be more authoritative even than the judgment published by the court itself. The ICLR’s website explains:
However important the case, the transcript of a judgment does not have the same value, as a record of the decision contained in it, as a full text law report. Where, therefore, a law report is available, particularly where the case has been reported in one of the official series, The Law Reports, published by ICLR, it must be cited and referred to in that version in preference for any other: see Practice Direction (Citation of Authorities)  1 WLR 780 … the combination of a summary report and a transcript does not enjoy the same status as a full text law report where one is available, for reasons which will become clear once one appreciates the amount of careful editorial work that goes into preparing the full report.
In other jurisdictions, the official law reports are treated with less reverence, although it is still normal for courts to require them to be cited in preference to primary sources such as slip opinions. Nonetheless, further down the page the ICLR makes clear that the full text of the judgment takes primacy over the headnote, just as in other jurisdictions:
Although the headnote adds value to the report of a judgment, it is the judgment itself which sets the precedent and binds subsequent decision-makers.
Nevertheless, a good headnote is a work of meticulous legal draftsmanship and can withstand the closest of textual scrutiny. Accuracy is all. Make the proposition too general, and the principle appears either so watered down as to be meaningless or, more dangerously, misleadingly wide in its application. Draw it too narrowly, and it fails to express the principle on which the court based its decision. At its best, a headnote is a precious distillation – the single malt of legal learning.